Closed plus finite dimensional in a TVS If $E$ is a topological vector space (TVS), $F_1$ a closed subspace of $E$, and $F_2$ a finite dimensional subspace of $E$, such that $F_1 \cap F_2=\{0\}$, is $F_1+F_2$ necessarily closed? If yes, are the projection from $F_1+F_2$ onto $F_1$ and $F_2$ respectively, continuous? 

Whenever N,F are subspaces of a TVS, N closed and F finite-dimensional, N+F is closed.
This is Theorem 1.42 (p. 30) in Rudin: Functional Analysis, 2nd edition, 1991.
https://archive.org/details/RudinW.FunctionalAnalysis2e1991
But even if the TVS is Hilbert, F being closed is not enough (By Exercise 1.20, p. 40), it must be finite-dimensional.
Theorem 5.16b (p. 126):
If N and F are closed subspaces of an F-space N+F, and $N\cap F=\{0\}$, then the projection $P:A+B\to A$ with null space $B$ is continuous.
So to your first question the answer is "yes", to the second "yes if N+F is an F-space" but not in general, by Exercise 5.9.
Exercise 5.9 (p. 145): Here $N,F$ are closed subspaces over $L^2(0,1)$, $N\cap F=\{0\}$ but yet $P$ is not continuous.
Note: it is not enough that N,F are closed subspaces of the F-space $L^2$; the space $N+F$ should be complete (F-space).
Def. 1.8e: F-space = TVS whose topology is induced by a complete invariant metric (equivalently, TVS having a complete N1 topology, by Theorem 1.24)
BTW, only in Theorem 1.42 finite-dimensionality is assumed, not in the others. 
In Rudin, TVSs are Hausdorff.
