Clopen subsets of tdlc groups

No. For instance, let $G=\mathbb{Z}_p\times\mathbb{Z}$ and consider the set $U=\bigcup_{n\in\mathbb{N}} p^n\mathbb{Z}_p\times \{n\}$. More generally, given any non-discrete totally disconnected locally compact group $H$, you can take a product of $H$ with a discrete group and then take a union of shrinking clopen subsets of different cosets of $H$ in the product.

I think the answer is no.

For example, let $G$ be your favorite non-discrete tdlc group such that there exists $U_0\supseteq U_1\supseteq \dots$ a decreasing sequence of clopen neighborhoods of the identity, with $\bigcap_{n=0}^\infty U_n = \{e\}$, but $U_n\neq \{e\}$ for all $n$. For example, you could take $G$ to be the automorphism group of the complete binary tree and $U_n$ to be the set of all automorphisms which fix the ball of radius $n$ around the root.

Now consider $G\times \mathbb{Z}$, which is also tdlc. Let $U = \{(x,n)\mid n\geq 0, x\in U_n\}$. This set is clopen in $G\times \mathbb{Z}$. But for any neighborhood $V$ of the identity $(e,0)$, there is some $n$ such that $V\not\subseteq U_n$. So letting $g = (e,n)$, we have $gV\not\subseteq U$.