Traxler is just not very wise against all but the most inexperienced players. That being said, an element of it that I do actually like is when someone tries the fried liver attack on me and eagerly(and stupidly in my opinion) wants to trade their bishop and knight for my rook and pawn(after castling).
Is the Traxler Counter Attack a solid opening?

But this doesn’t necessarily mean anything when humans play. The white king is going to be chased around and has to defend perfectly in order to not get into trouble.
So if white survives the initial onslaught, black is in trouble. It definitely makes for a fun game for black, especially if you like attacking.

I never play the fried liver (I play king's gambit instead) so I've never had to deal with the Traxler Counterattack.
But if I did, I'd probably decline the free bishop. I'd play something like this:

Neither solid, nor sound- unless white is either poorly booked, or an idiot, and plays 5.Nxf7.
Why is 5. Nxf7 bad? Is it because you think 5. Bxf7+ is better?

The original question was if the Traxler is solid, and it certainly doesn't qualify as solid. "Double-edged" would be a better word for the opening. At the club level, especially with scholastic players, I think the Traxler has good psychological value - players who play 4. Ng5 like to attack, and 4...Bc5!? turns the tables on them, forcing them to think about playing defense. So against certain players the Traxler can be very effective, although you won't find many GM games with it.

Neither solid, nor sound- unless white is either poorly booked, or an idiot, and plays 5.Nxf7.
Why is 5. Nxf7 bad? Is it because you think 5. Bxf7+ is better?
5.Nxf7? leads to an equal position with best play from both sides- while 5.Bxf7+ as well as 5.d4! give white a large advantage.
And I do not think anything, the above are well established facts.

Nothing interesting here- a 36 yr old game with inaccurate moves by both sides ( e.g. 6...Rf8?! should be answered by 7.0-0!), and then 11.f3 is suicidal.

The Traxler pretty much breaks everything that defines a solid opening. But it's not necessarily bad, especially at your level.

This is greedy white
Or a game between idiots- whatever comes second.
After 7...Qf6?? either 8.Rf1 or 8.Qf3 win outright.
Traxler is not sound, but it is occasionally played at grandmaster level as a surprise
https://www.chessgames.com/perl/chessgame?gid=1018213
Besides 4 Ng5 is not considered sound either and grandmasters nowadays prefer the tame 4 d3.

This is greedy white
Or a game between idiots- whatever comes second.
After 7...Qf6?? either 8.Rf1 or 8.Qf3 win outright. So it's all the same

Traxler is not sound, but it is occasionally played at grandmaster level as a surprise
https://www.chessgames.com/perl/chessgame?gid=1018213
Besides 4 Ng5 is not considered sound either and grandmasters nowadays prefer the tame 4 d3.
There is no surprise here- Anand was winning throughout the game, missed a couple of trivial wins... and blundered badly at a winning position with his very last move.
#18
Maybe Anand burned too much time and that explains it.
Here is another one of GM beliavsky.
https://www.chessgames.com/perl/chessgame?gid=1068378

#18
Maybe Anand burned too much time and that explains it.
Here is another one of GM beliavsky.
https://www.chessgames.com/perl/chessgame?gid=1068378
Not very likely. Anand used to play fast, and the blunder comes right after the first time control, so he must have had more than enough time. I think an unlucky moment.
The second one is a bit suspect. Karpov and Beliavsky were teammates at the Soviet National Team, and to my knowledge this is the one and only game that Karpov played 4.Ng5 in his life.
There is just something very tempting about the traxler counter attack that I just find very hard to resist. When it works it gives a great feeling however I suspect it will work only against those who don't know how to deal with the bxf2 sack.
If you play Kf1 instead of taking then the sting is lost and black is clearly worse. However when I play it, it usually has delivered the results.
However is traxler counter-attack a solid opening.