1. e4 d5 2. e5!!?

Then you are playing some weak opposition that has no understanding of chess.

The problem with the French is the bad Bishop

The problem with the Caro-Kann is it's slowness. Gotta play c6 to defend d5 in order to get the Bishop out, then play e6 to switch the role of which pawn covers d5, and then move that c-pawn again to attack d4.

But after 1.e4 d5 2.e5?, Black gets the best of both worlds. He gets his bad Bishop outside the pawn chain, like in the Caro, combined with ...c7-c5 in one go, immediately attacking White's center, like in the French.

I played it in chess tournaments. I am used to the positions. At higher levels I have no doubt it is weaker but Scandinavian players like open positions. If you close up the center I find that many get frustrated.

That's all I am saying

It's really not alot like the typical advanced french, you can't even castle queenside / push your kingside pawns due to the bishop pressuring the queenside / guarding the diagonal from attack (the typical advanced french plan). Pushing f4, a typical move in the french, leads to a -.7 swing in the engine eval. I think you just have to castle and play a very normal game that's equal.

Forgetting about the engine eval... which says it's equal, right around 0.00. There are some good openings for white that end up scoring around equal. After pushing e5 the moves just seem to flow completely naturally for black. There is nothing to do in the position, there is no complexity or pressure.

If you really want to throw the opponent off in the Scandinavian this is much better... there are some very surprising kingside attacks here. Dubious at master level but anywhere below that it scores very well. You can also transpose non-scandinavian players into it via 1. Nc3 d5 2. e4 which will make it alot more reliable at higher elo -

The problem is that after 4...Be6! white has to lose time to put the Bf1 to a useful square (Bb5+/Ba4/Bb3) and Black will have time to consolidate a useful space advantage.

I know, this is just a risk you accept when you play the line. But if they don't play 4. Be6 (only played 25% of the time at 2200+ on lichess) you have some very strong kingside attacks with 60% winrates, so you can argue that statistically the risk pays off. But this is why I like it transposed from 1. Nc3 rather than from 1. e4 - some Scandinavian players will respond correctly... but I doubt 1. Nc3 d5 players are going to.

Anyway, if they play Be6 it's not lost, you're just going to need to play very sharpy for a while. But you can do that, at least you know what your goal is. Black still has to play sharply for a while to maintain the engine advantage as well. For example... 5. c3 and at master level the most common move is 5... c5, which is wrong and after 6. Bb5 Nc6 7. Nf3 the game is just about equal. But yes if you want to play good engine moves you have to play the mainline Scandinavian.

But I would choose this Nc3 option 100% of the time over pushing 2. e5. Even at master level Nc3 still scores even... at master level pushing e5 black wins about 50% of games and white wins 15%.

a : The French Exchange Defense and other such openings are called equal and imo it is an error to believe they would necessarily be draws. They are equal in the sense that there is no a really advantage and therefore White and Black start with equal luck which doesn't necessarily mean draws but also wins for White or Black.

This is to say that after 1) e4 d5 2) e5 Bf5 I think it's an equal opening and that for example Magnus Carlsen imo would be happy to play this opening with Black because he had a position with equal luck but if he plays against Stockfish in this equal opening, it is likely that he will still lose.

b : 1) e4 d5 2) exd5 Nf6 3) c4 e6 (also 3...c6 is annoying) is a dangerous gambit that can annoy and maybe lead to playing the equal opening of the topic but it should be noted that this opening choice is quite weak because even if Black plays more passive than 2)...Bf5 i.e 2)...e6 3) d4 c5 we just transposed into a French advance which is not so much bad news for Black (and the story that White would have an advantage because of the bad Bishop c8 is rather a joke to me ex. 4. c3 Qb6 5. Nf3 Bd7 6. Be2 cxd4 7. cxd4 Bb5)

c : To avoid the Icelandic gambit I prefer to play simply 1) e4 d5 2) exd5 Nf6 3) d4 Nxd5 and I will still play a game with a space advantage which is still better than the opening of the topic imo.

Well you can insist that if it makes you feel good inside but we have the data from lichess and from masters level games, and you're just simply wrong according both datasets - infact at 2200+ on lichess rapid it has a higher winrate than the mainline scandinavian - and that's mostly scandinavian players, I'm talking about transposing people into it which looks like it adds about 5% to the winrate beyond that, so... no, wrong, the end.

I ran into this weird position earlier...

Recognize it? It is this same improved french we've been talking about... except white is playing the french this time, and the position is the mainline of the Saragossa opening. 
I would actually be happy to play white here, and the engine does slightly prefer white despite the repeated pawn move ... for the reasons we've discussed.

It even transposes in the QGD exchange...

The masters database is not American-based, I'm speaking to the data, not my personal anecdotes. That should be obvious but apparently with you it needs pointing out.

No, keep trying