1) Consider the semidirect product $G=\mathbf{Z}^2\rtimes\mathbf{Z}$ with action by the matrix $A=\begin{pmatrix}2 & 1 \\ 1 & 1\end{pmatrix}$. It has the presentation
$$\langle t,x,y\mid txt^{-1}=x^2y,\; tyt^{-1}=xy,\; [x,y]=1\rangle.$$
(It is actually generated by $(t,x)$).

Claim: some limit of markings on $G$ fails to be noetherian (although $G$ is)

[Added: this result was proved by Luc Guyot (*Limits of metabelian groups. Internat. J. Algebra Comput. 22 (2012), no. 4, 1250031*; ArXiv link, MR link), showing (2nd item of Theorem A therein) that suitable markings accumulate to the free metabelian group on 2 generators.]

*Proof of the claim:* [Edit: It's enough for purposes here to prove something weaker than Guyot's theorem, namely showing that some limit admits $\mathbf{Z}\wr\mathbf{Z}$ as a subgroup.] Consider the generating subset $(t^{n^2+1},t^n,x)$. There is a unique homomorphism $j_n$ from the wreath product $\langle s,x\mid [s^mxs^{-m}, x]=1,\forall m \in \mathbb{Z}\rangle$ to $G$ mapping $s$ to $s:=t^n$ and $x$ to $x$. We claim that it is injective in a ball of radius $p_n$ going to infinity with $n$. Once this is proved, this shows that some limit of copies of $G$ (any accumulation point for this choice of marking) contains a copy of $\mathbf{Z}\wr\mathbf{Z}$ and hence is not noetherian.

To prove the claim: assume the contrary. This means that some nontrivial element $w$ of the wreath product such that $j_n(w)=1$ for infinitely many $n$ (say $n\in I$, infinite set of integers). Clearly $w$ should be in the kernel of the canonical map to $\mathbf{Z}$, and therefore can be written $w=P(x)$ with $P\in\mathbf{Z}[u^{\pm 1}]$ (Laurent polynomial ring). Here for $P=\sum a_nu^n$, $P(x)$ means $\prod s^nx^{a_n}s^{-n}$. (This is a standard identification, since $\mathbf{Z}\wr\mathbf{Z}\simeq\mathbf{Z}[u^{\pm 1}]\rtimes\mathbf{Z}$).

That $j_n(w)=1$ means that, in $\mathbf{Z}^2$, $P(A^n)x=0$. Since $A$ is irreducible, that $P(A^n)$ has nontrivial kernel implies that it's zero. So for all $n\in I$, we have $P(A^n)=0$. But the $A^n$ have distinct eigenvalues (except for $n$ and $-n$), so infinitely of them cannot be killed by a single polynomial, contradiction.

Note that $G$ is linear over $\mathbf{Z}$, hence equationally Noetherian.

2) Small groups. Consider, for instance, a f.g. group $H$ containing all finite groups as subgroups (e.g., Neumann's group $\mathrm{Sym}_0(\mathbf{Z})\rtimes\mathbf{Z}$, or its finitely presented analogue due to Houghton), with no free subgroup.

Claim: some limit of markings on $H$ contains a free subgroup (although $H$ doesn't)

Fix a sequence of finite marked groups $(V_n,x_n,y_n)$ tending to the free groups on 2 generators. Then consider a sequence of generating subset of $H$, as a concatenation of a fixed generating subset of $H$, and a pair $(x'_n,y'_n)$ in $H$ generating a subgroup isomorphic to $(V_n,x_n,y_n)$ as marked group. Then any accumulation point of this sequence of marked groups has the property that the last two generators freely generated a free group.