= Solution
Put $U_0=G$ and $U_{j+1}=U_j^p$, where powers mean closed generated subgroups. The finite-quotient arguments of Question 3 pass to a <powerful pro-p group>: in particular $U_j$ is <powerfully embedded>, the power map is onto $U_{j+1}$, and
$$
[U_j,G]\subseteq U_{j+1},\qquad (xu)^p\equiv x^pu^p\pmod {U_{j+2}}\quad(u\in U_j).
$$
For the root assertion, apply the finite result to every finite quotient and use the <finite intersection property> on the closed sets of possible roots in the <compact group>. Each $U_j$ is finitely generated by the appropriate powers of a finite generating tuple, and $U_j/U_{j+1}$ is finite and <elementary abelian>. Thus the $U_j$ are open. They form an identity-neighbourhood basis: in any finite continuous quotient their images form a terminating <lower p-series>, so some $U_j$ is contained in the kernel. In particular $\bigcap_jU_j=1$. Collection also gives
$$
[U_a,U_b]\subseteq U_{a+b+1}.
$$
One can see the extra index directly by induction from $[U_a,G]\subseteq U_{a+1}$: collect $[v,w^p]$ for $w\in U_b$. Its leading factor $[v,w]^p$ advances one layer, and the remaining iterated <commutators>, with their binomial coefficients, lie at least as deep. Products and limits preserve the inclusion. This is the usual strongly central property of the <lower p-series>.
A <uniform pro-p group> is a finitely generated <powerful pro-p group> for which all the indices $|U_j:U_{j+1}|$ equal $|G:G^p|$. The preliminary congruence gives surjective linear maps
$$
\theta_j:U_j/U_{j+1}\longrightarrow U_{j+1}/U_{j+2},\qquad uU_{j+1}\longmapsto u^pU_{j+2}.
$$
If $G$ is uniform, these maps are bijective. If $x\ne y$, write $y=xu$ and choose the unique layer $j$ with $u\in U_j\setminus U_{j+1}$. Bijectivity of $\theta_j$ and the power congruence imply $y^p(x^p)^{-1}\in U_{j+1}\setminus U_{j+2}$, so $x^p\ne y^p$. Thus the power map is injective. A finite-order element of a <pro-p group> has $p$-power order; repeated injectivity of the power map excludes every nonidentity such element. Hence a uniform group is a <torsion-free group>.
Conversely, suppose some $\theta_j$ has a nonzero kernel. Choose $x\in U_j\setminus U_{j+1}$ with $x^p\in U_{j+2}$. At step $r\geq j+1$, if the current $x_r$ satisfies $x_r^p\in U_{r+1}$, surjectivity of $\theta_r$ supplies $u_r\in U_r$ with
$$
u_r^p\equiv(x_r^p)^{-1}\pmod {U_{r+2}}.
$$
Set $x_{r+1}=x_ru_r$. The power congruence gives $x_{r+1}^p\in U_{r+2}$. Since the corrections tend to $1$ in the complete <profinite group>, the $x_r$ converge to $z$, with $z^p=1$ and $z\equiv x\pmod {U_{j+1}}$. In particular $z\ne1$. A <torsion-free group> cannot have such an element, so every $\theta_j$ is injective and all layer indices are equal. This proves the <root correction criterion for uniformity> and
$$
\boxed{G\text{ is uniform}\ \Longleftrightarrow\ G\text{ is torsion-free}.}
$$
For a uniform group the power maps are continuous bijections $U_j\to U_{j+1}$, hence <homeomorphisms>. Their inverses define the unique roots used below. Comparison at the first differing layer also proves the <power congruences in a uniform pro-p group>:
$$
x\equiv y\pmod {U_b}\quad\Longleftrightarrow\quad x^{p^a}\equiv y^{p^a}\pmod {U_{a+b}}.
$$
The <intrinsic Lie algebra of a uniform pro-p group> has the same underlying set as $G$, zero equal to the group identity, and operations
$$
x+_Ly=\lim_{n\to\infty}(x^{p^n}y^{p^n})^{p^{-n}},\qquad
[x,y]_L=\lim_{n\to\infty}[x^{p^n},y^{p^n}]^{p^{-2n}},\qquad
\lambda x=x^\lambda\quad(\lambda\in\mathbb Z_p).
$$
Here $x^\lambda$ is the limit of integral powers approaching $\lambda$. The first root is taken in $U_n$ and the second in $U_{2n}$; the commutator actually lies in $U_{2n+1}$. The following faithful algebra realization verifies convergence and the <Lie algebra> axioms, rather than assuming them as part of the definition.
Choose a minimal tuple $a_1,\ldots,a_d$ of <topological generators>. The layer isomorphisms $\theta_j$ show that the images of $a_i^{p^j}$ are bases of $U_j/U_{j+1}$. The ordered-coordinate argument of Question 3 therefore gives a continuous bijection
$$
\mathbb Z_p^d\longrightarrow G,\qquad (\lambda_i)\longmapsto a_1^{\lambda_1}\cdots a_d^{\lambda_d}.
$$
Existence follows by successive digits and <compactness>; uniqueness follows by the first digit at which the coordinates differ, because its images are linearly independent in the corresponding power layer. Thus this map is a <homeomorphism>.
Here are details of the norm needed on the <group algebra>. In $\mathbb Z_p[[G]]$ put $b_i=a_i-1$. Its elements have unique ordered expansions $\sum_\alpha c_\alpha b_1^{\alpha_1}\cdots b_d^{\alpha_d}$ with $c_\alpha\in\mathbb Z_p$. To verify this description, identify the ordered coordinates of $G$ with $\mathbb Z_p^d$. Elements of the <completed group algebra> are integral measures on this space; their values on the binomial polynomials $\prod_i\binom{\lambda_i}{\alpha_i}$ are precisely the coefficients $c_\alpha$. The finite-difference identity pairs the ordered monomial $b^\beta$ with this binomial polynomial as $\delta_{\alpha\beta}$. The <Mahler expansion> of a continuous function therefore proves both existence and uniqueness of the measure with the specified coefficients. This also proves that the ordinary rational <group algebra> embeds in these expansions.
Choose $1/(p-1)<s<1$ and set
$$
v\left(\sum_\alpha c_\alpha b^\alpha\right)=\inf_\alpha\{v_p(c_\alpha)+s|\alpha|\}.
$$
This defines the <weighted group algebra of a uniform pro-p group>. Its multiplication is submultiplicative in the valuation convention, $v(AB)\geq v(A)+v(B)$. Indeed, $a_i^p-1=pb_i+\binom p2b_i^2+\cdots+b_i^p$ has valuation $1+s$; collection of ordered coordinates gives the same lower bound for $g-1$ when $g\in U_1$. Interchanging two adjacent $b_i$ in a product introduces a correction coming from $[a_i,a_j]-1$, of valuation at least $1+s>2s$. Iterating the interchange rule never lowers the total weight; the corrections have strictly increasing weight, so infinite collection converges. The same argument with $p^j$-th powers shows
$$
v(g-1)=s+j\quad\text{for }g\in U_j\setminus U_{j+1}.
$$
The equality follows from the first nonzero coordinate digit: its leading terms are $p^j\lambda_i' b_i$, at least one with unit $\lambda_i'$, and cannot cancel in the ordered expansion. Completing the rational <group algebra> in this norm produces a complete <normed algebra> $A$ in which $G$ embeds faithfully and with its original topology.
Since $v(g-1)\geq s>1/(p-1)$, the series
$$
\log g=\sum_{r\geq1}\frac{(-1)^{r+1}(g-1)^r}{r},\qquad
\exp X=\sum_{r\geq0}\frac{X^r}{r!}
$$
converge on the relevant balls in $A$ and are inverse there. Their convergence follows from $v_p(r!)\leq(r-1)/(p-1)$ and $v_p(r)\leq(r-1)/(p-1)$; formal one-variable identities prove that they are inverse. Put $X=\log x$ and $Y=\log y$. The <Baker--Campbell--Hausdorff formula> in $A$ now gives
$$
p^{-n}\log(x^{p^n}y^{p^n})\longrightarrow X+Y,
$$
because every term of degree at least two tends to zero after this rescaling. Similarly expansion of $\exp(-p^nX)\exp(-p^nY)\exp(p^nX)\exp(p^nY)$ gives
$$
p^{-2n}\log([x^{p^n},y^{p^n}])\longrightarrow XY-YX.
$$
The leading term is the indicated associative-algebra commutator and all terms of degree at least three tend to zero. These expressions are the logarithms of the unique roots in the intrinsic formulas. The image $\log G$ is compact and hence closed in $A$, so both limits belong to $\log G$, and the inverse logarithm takes the limits back to $G$. Also $\log(x^\lambda)=\lambda\log x$. Consequently $\log$ identifies the intrinsic operations with addition, scalar multiplication and commutation in $A$. The <Jacobi identity> follows by expanding the associative products; thus $L_G$ is indeed a <Lie algebra> over $\mathbb Z_p$.
The commutator roots in the intrinsic formula belong to $U_1$, so their limit gives $[L_G,L_G]\subseteq\log U_1=pL_G$. Iterating the power congruence gives $x^{p^n}y^{p^n}\equiv(xy)^{p^n}\pmod {U_{n+1}}$. Taking the unique roots shows that intrinsic addition modulo $U_1$ is the original group product modulo $U_1$. Consequently $L_G/pL_G$ is the original elementary abelian quotient $G/U_1$, and the images of the $a_i$ form a basis. Successively removing digits expresses every element uniquely as an additive sum $\sum_i\lambda_i a_i$: existence uses completeness, while a nonzero relation would have a first nonzero digit contradicting the basis in $U_j/U_{j+1}$. Hence $L_G$ is a <free module> of rank $d$ over the <p-adic integers> and is a <powerful Lie lattice>.
Conversely a <powerful Lie algebra over p-adic integers>, for odd $p$, is a finite free $\mathbb Z_p$-module $L$ with a bilinear alternating <Lie bracket> satisfying the <Jacobi identity> and $[L,L]\subseteq pL$. Define
$$
X*Y=\operatorname{BCH}(X,Y)
=X+Y+\tfrac12[X,Y]+\tfrac1{12}([X,[X,Y]]+[Y,[Y,X]])+\cdots.
$$
For completeness, integrality and convergence can be verified without guessing which individual denominators are units. In $\mathbb Q_p\otimes L$ set $M=p^{-1}L$. Then $[M,M]\subseteq M$, so its integral <universal enveloping algebra> has an ordered basis by the <Poincaré-Birkhoff-Witt theorem>. Its coefficientwise $p$-adic norm is submultiplicative, and $L=pM$ consists of elements of valuation at least one. The homogeneous degree-$r$ terms in $\log(\exp X\exp Y)$ have coefficients of valuation at least $-\lfloor(r-1)/(p-1)\rfloor$: expand the logarithm, use $v_p(k!)\leq(k-1)/(p-1)$ on each nonconstant exponential factor, and $v_p(\ell)\leq(\ell-1)/(p-1)$ on a product of $\ell$ such factors. Therefore evaluating degree $r$ on $X,Y\in pM$ gives valuation at least
$$
r-\left\lfloor\frac{r-1}{p-1}\right\rfloor,
$$
which tends to infinity and is at least two when $r\geq2$. Each homogeneous term is a rational <Lie polynomial> by the formal <Baker--Campbell--Hausdorff formula>. The ordered enveloping-algebra basis shows that its intersection with $\mathbb Q_pM$ is exactly $M$, so the estimate places the sum in $L$, with the non-linear part in $pL$.
Associativity follows from the identity $\exp(X*Y)=\exp X\exp Y$ in the completed <universal enveloping algebra>. Its identity is $0$, its inverse is $-X$, and $X^{*m}=mX$, since the bracket of $X$ with itself is zero. The same coefficient estimate for the group commutator puts it in $pL$. Conjugation in the BCH group is $\exp(\operatorname{ad}X)$; each term $\operatorname{ad}(X)^r/r!$ preserves $p^jL$, because each bracket adds a factor of $p$ and $v_p(r!)<r$. Thus the submodules $p^jL$ are normal subgroups for $*$, their intersection is zero, and their finite quotients have $p$-power order. Thus $(L,*)$ is a <powerful pro-p group>, with
$$
(L,*)^{p^j}=p^jL,\qquad |p^jL:p^{j+1}L|=p^{\operatorname{rank}L}.
$$
A module basis topologically generates this group: it generates the quotient $L/pL$, and successive power digits lift that generation through all $p^jL$. \b[The BCH construction therefore gives a uniform pro-p group.]
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