I am kinda torn between playing either the Tarrasch (Nd2) or Nc3 (Winawer?) against the French. What are some advantages and disadvantages to each of those two approaches?

Also, are there any grandmasters that play 1 e3? It would be interesting to see some recent 1 e3 games.

It's an uncommon opening and isn't seen played much by GMs. It's been used as a (relatively) flexible move to slip into a more conventional system.. And there are a ton of different systems to pick from which include an early e3.

Nimzowitsch played it a couple of times. He drew against Yates transposing into what is either a Nimzo-Larsen, Bird or a sort of reversed Dutch system; with the same system he beat Euwe.
He lost to Vidmar going into what I would call a reversed QID. He won against Tarrasch who went for what is probably the unique line of the van 't Kruijs, 1. e3 e5 2. Nc3 d5 3. d4 exd4 4. Qxd4, but I believe Tarrasch could've played the endgame better. There are a few games where he comes up with something funny-looking (a line with e3-c3-d4-b4-a4-a5) including a nice win over Spielman...
I remember Blatny lost a game in the World Open in 2003 or 2004, that's all I can think of.

Question 1: The advantage of nc3 is that it is the best square for the knight, and influences the center. However, it blocks the c-pawn and can be pinned by Bb4.

Nd2 is more flexible, as the knight can go to f3 if the other knight goes to e2. Seems a little strange at first, but I've seen it done. Also, no pins, as the c-pawn can intervene. The downside of Nd2 is that it blocks the c-bishop, and does not directly influence the center. However, a pawn on c3 helps solidify the center.

Question 2: 1. e3 is not very popular in GM play or at the club level, but I'm sure some will use it. It tends to steer the game towards reversed French (1. e3 e5 2. d4), reversed Nimzo-Indian, the Nimzo-Larsen attack (1. e3 d5 2. Nf3 c5 3. b3), or some form of the flexible Reti Opening if black does not play 1. ...e5.

Something to try, I suppose. Probably not reliable enough to be a main opening as White, but definitely lower on theory than more mainstream openings like e4 or d4.

It's a fair enough way to open a chess game, you'll be equal pretty much against any solid black defense though. As for Nc3, it's a good square for the knight, posting it somewhere else is entirely possible but not a drop better (or worse).

I remember playing this in a blitz game a looong time ago: 1. e3 f5 2. b3 Nf6 3. Bb2 e6 4. f4 Be7 5. Be2 0-0 6. Nf3 b6 7. 0-0 Bb7... A very interesting game that I won later on.