What dynamic openings for Black are considered sound against 1.e4?
A few openings come to mind immediately come to mind and aren't to "theory heavy" for beginners. Two that I use to frequently come up against between the 800 - 1500 ELO rated players were the Fried Liver Attack which you can convert to the Traxler Counter Attack. The Other was a fairly central game from white which I find the Caro Kann quite effective against. I'll add links to the chess.com opening books
The Caro Kann is quite popular but very positional in my opinion. Also probably the most theory based. Just learn the exchange variation. Its super solid and give a good foundation. The game usually begins to open up quite a lot in the middle game and I find that very dynamic positions are easily achieved.
Theory: Probably one of the lest dynamic openings for black initially. I usually find the positions open up in the middle game and end up in really interesting and dynamic positions. It also usually leads to really good end games too. I strongly suggest this one because it helps with the development of tactics and strategies better than most black opening in my opinion.
There are a lot of lines to learn but ultimately they all end up with a d7-d5 push in the stronger one.
[FEN ""] 1. e4 c6 2. d4 d5 *
The Sicilian Defence probably the most theory dense opening in existence. It is quite a good opening in my opinion but if you really want to get into there are so many lines it just hurts my head .
Theory: c5 completely unbalances the board from the get go. You'll often find that whites pieces gravitate more to the queen side in the opening. Strongly suggest looking into the Open Variation as its the least painful to remember.
[FEN ""] 1. e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 *
The Petrov Defence another nice opening. Allows for rapid development is very sound opening , Reasonable defence options. You'll find you will often end up being able to play a Stafford Gambit which is pretty fun but would also suggest against it as though its a "trappy" opening, its really easy to defend against.
[Event "Petrov Defence"] [White "White"] [Black "Petrov Defence"] [FEN ""] 1. e4 e5 2. Nf3 Nf6 *
There is another which is the Kings Indian Defence. I really don't know a lot about it though sorry.
The Traxler Counter Attack is quite a poisonous little game. For some reason 800-1500 love playing fried liver games looking for the quick win this opening I find is really good for giving them a whack around the ears. I generally wouldn't recommend this one though. It is great at lower levels but fairly useless from intermediate levels onwards. It isn't necessarily seen as a "Sound" opening by most however the mainlines show a fairly equal position around move 7 with Black having significant counterplay.
Theory: The idea is that you are not concerned about the Nxf7 because you can immediately take back the f2 pawn , checking the king. The king usually takes the bishop and you immediately follow up with Nxe4+ the king has a huge number of squares it can go to however ... only 2 of them don't result in significant loss of material or being mated. Levi Rozmann has a good video on the Traxler off memory.
[FEN ""] 1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. Ng5 Bc5 5. Nxf7 Bxf2+ *
Post Scriptum: Ultimately openings aren't that beneficial at a beginner level. You are far better off focusing on tactics and pattern recognition. Openings are also extremely subjective , eg; I think the Caro Kann is dynamic however the majority of players would think the opposite. I hate playing the Traxler, just doesn't seem to sporting in my opinion. Here are a few to look into and just try a few out , extend the theory and understanding.
Personally I find I user maybe 3 openings as White and 4 as Black, and this has basically managed to get me through 90% of my tournaments.
White: Italian Game, English Opening, Kings Fiancheto
Black: Sicilian Defense, Kings Pawn - Kings Knight variation, Petrov Defence , Caro Kann