An algebra $A$ has the congruence extension property (CEP) if for every $B\le A$ and $\theta\in\operatorname{Con}(B)$ there is a $\varphi\in\operatorname{Con}(A)$ such that $\theta =\varphi\cap(B\times B)$. A class $K$ of algebras has the CEP if every algebra in the class has the CEP.
On
If a lattice $L$ has the pentagon $N_5$ as a sublattice ($N_5=\{0,a,b,c,1\}$, with $0<a<b<1$, $0<c<1$, $a\wedge c=b\wedge c=0$ and $a\vee c=b\vee c=1$), then $L$ doesn't have the CEP, because the chain $\{0,a,b,1\}$ is a sublattice where $$\theta(0,a)=\{0,a\}^2\cup\{(b,b),(1,1)\},$$ but in $N_5$, if $0\theta a$ then $c\theta 1$, and so $0\theta b$, and so $$\theta_L(0,a)\supseteq\{0,a,b\}^2.$$ So there is no congruence of $L$ that restricts to $\theta(0,a)$ in $\{0,a,b,1\}$.
This together with the answer of Arturo Magidin shows that a lattice has the CEP iff it is distributive.
The variety of all groups does not have the property. Congruences correspond to normal subgroups, so you are essentially asking whether if $H\leq G$, and $N\triangleleft H$, does there always exist an $M\triangleleft G$ such that $M\cap H= N$. This does not hold; for example, if $G=A_5$, $H=A_4$, and $N$ is a nontrivial proper normal subgroup of $A_4$, then you cannot find any appropriate $M$.
The variety of all latices does not have the property either. Let $L$ be the nondistributive lattice $M_3$, with elements $0$, $1$, $x$, $y$, and $z$ (where the join of any distinct elements of $\{x,y,z\}$ is $1$, and the meet is $0$). Let $M$ be the sublattice $\{0,x,1\}$. Then let $\Phi$ be the congruence in $M$ that identifies $0$ and $x$.
Let $\Psi$ be a congruence on $L$ that identifies $0$ and $x$. Then it must identify $0\vee y=y$ with $x\vee y=1$; similarly, it must identify $0\vee z = z$ with $x\vee z = 1$. Thus, $y$, $z$, and $1$ are identified in $\Psi$. That means that $z\wedge 1 = z$ must be identified with $z\wedge y = 0$, so all of $0$, $z$, $y$, and $1$ are identified in $\Psi$. That means that $x$ is identified with $1$ as well, so that $\Phi$ is a proper subcongruence of $\Psi|_M$. Thus, $M_3$ does not have the congruence extension property.
In fact, I believe (but don’t have access to my textbooks right now) that the Congruence Extension Property precisely characterizes the distributive lattices among all latices (see for instance the opening line of this paper ) which would give an affirmative answer to your final question.