4.Ne2 is well playable, and has 500 times less things to learn than the mainlines.

There is not accumulated Ne2 theory in any repertoire book, afaik. You can find an interesting article in "Secrets of Opening Surprises 12" named "The French a la Hector" by Israeli GM Alex Finkel.

THis isn't a serious issue. Simply 6.b4 Bb6 7.Nxe4, and the question is if the bishop on b6 is pressuring d4, or it's factly a fat pawn. I am inclined towards the latter view.

5.a3 Be7 is surely the most testing option, when Black is rather comfortable if white gets the pawn back immediately, but Hector's 6.g4 (aiming at displacing the f6 knight before taking back the pawn) is IMO the critical test.

Taking on c3 is another option. 5.a3 Bxc3+ 6.Nxc3 Nc6, when 7.Bb5 is enterprising but probably fails short, but the "dull" 7.d5 ed5 8.Qxd5 is probably equal with best play, but Black must be extremely careful.

The Ne2 line is one of the lines I am recommednding to my "advanced" students. To the newbies, I suggest just 2.d4 d5 3.Bd3 (equal, but with reasonable play for both sides, and not dull), and to the tactically aware ones, the lesser Steinitz 2.Nf3 d5 3.Nc3 Nf6 4.e5 Nfd7 5.d4 c5 6.dc5

I am playing myself Winawer/Steinitz proper, but this is way too much theory, and also very sharp: One subtle mistake, either colour, and bang... you are dead.

7.h3 ed4 8.Qxd4 when 8...Qxd4 is 100% better for white, and 8...Nc6 9.Qxe4!? messy, but most probably also good for white. That leaves 7...h5! which virtually forces white to sacrifice the g4 pawn via 8.g5 Bxg5 9.Bxg5 Qxg5 10.Rg1 Qh4 11.Qd2 or 11.Nd5, which may be dynamically balanced, but almost certainly not "dead even". Especially the crazy line 11.Nd5 Qd8(!) 12.de5 Nc6 13.Nef4! Nge7 14.Rxg7 Kf8 15.Rg3 Nf5 16.Rc3 Nxe5 17.Qe2! is actually closer to "hair raising" than "dead even".