In the first example 3.e4 is not good. White should not play that setup if Black's dark-squared bishop can be developed to c5. 3...Bc5 looks good then.
4.Nd5 and White is already panicking! After a simple 4...Bc5 not only is White's knight vulnerable to c7-c6 but he has to take time out to protect his e-pawn that he just weakened. White will soon lose even more time with his White knight and Black should have a really good game here. White is simply ignoring opening principles and creating weaknesess here so obviously things aren't going to go well.
In the second example there are a few things you can do to get an equal game. The easiest is probably 2...Nf6 when 3.d4 exd4 4.cxd4 d5 is a French exchange. You just play chess here. Don't blunder and you'll be fine.
Otherwise 2...Nf6 3.Nc3 Nc6 4.Nf3 Bb4 is fine. You get all your pieces out and White needs to figure out what to do. White doesn't get anything out of playing d4 here other than lots of trades which Black shouldn't be worried about anyway.
2...c5 is probably ok but I'm not sure. At any rate 3.d4 is a mistake because of 3...exd4 4.exd4 cxd4 5.Qxd4 Nc6 winning a tempo for nothing.
The easiest way to play the English is to either try what you play against 1.d4 or just develop as your pieces quickly as if you were playing 1.e4 e5.
3. Nf3 and if 3....e4? then 4. Ng5 good position with white. The most common is g3+Lg2 and usually black can do this too. Theory books...
In the second position if white plays dxc5 then you can recapture with the bishop, if you take with dxc4 you risk exchanging queens and losing the right to castle.
White is probably better after cxd5 but Black souldn't be playing d5 in the first place.
no, white should play dxc5 to threaten to exchange queens, then when the bishop takes c5 white can play cxd5 and the pawn is protected. (at least temporarily).
In the first game, White should play the more popular lines 3 Nf3 or 3 g3. AnthonyCG is correct that e2-e4 is only good in certain situations, and this isn't one of them.
In the second, the move 2 e3 is just lame. It isn't a blunder, but it really isn't a serious attempt at advantage, either. For instance 3 Nf3 Nc6 or 3 Nc3 Bb4 and Black is fine in either case. 3 d4 is inferior after the simple ...exd4 4 exd4 Bb4+ and Black is already somewhat better placed, as he is ready for ...0-0 and ...d5 right away, seizing the initiative.