(A) The first premiss gives you
$A \lor B\quad$ (a)
The latter and the second premiss gives you
$D \land \neg A$
and hence of course
$\neg A\quad$ (b).
Put together (a) and (b) and you get
which together with the third premiss gives us
$A \lor E\quad$ (c).
(b) and (c) now give us
Fill in the steps using modus ponens and disjunctive syllogism, massage it into your preferred way of laying out proofs, and we are done!
(B) In the immortal words of the Hitchhikers Guide to the Galaxy, Don't Panic! There really was nothing magic going on here in finding the proof -- a bit of calm strategic thinking will see you through with this kind of example.
You did the right thing in working from the end and asking "How can I get $E \lor F$?", and noting that since $F$ wasn't around, you need to prove $E$.
How are you going to get $E$? It only appears the third conditional premiss's consequent, $A \lor E$, so somehow you are going to need to prove the antecedent of the conditional $B$, and then (probably) also prove $\neg A$ so you can use disjunctive syllogism to extract $E$. And yippee, there's a $B$ and a $\neg A$ around in the other premisses. Let's see if we can get at them!
Starting from the beginning then, the first box writes itself. And then it was obvious that the only thing to do with the original second premiss is use modus ponens which gets us to the second box and then the third box wrote itself.
Pause for thought. $D$ isn't going to be useful, but (b) is what we want, but the only way we can use it right now is to put it together with (a) and use disjunctive syllogism to get $B$.
And now we are motoring ....