Working on that right now. Will update original post when done.
Help understanding a high-blunder game

It wasn't marked as a mistake or an inaccuracy, but it certainly didn't go as planned from my perspective.
Pretty much all of my planning involves the word "hope." Maybe I don't know enough tactics, or I am just plain bad, but basically I don't know how to win if my opponent doesn't blunder.
Hmm...What I could have done differently... I don't see any other immediately open weak points to attack... Maybe I could withdrawn the queen back to safety until I brought more pieces out?

Thanks for all of the advice so far.
Yeah, I probably could have taken my queen back to safety sooner. I usually don't like taking my queen out this early, but somehow I convinced myself that taking the queen out was the correct response to Petrov's defense, classical variation.
If white had castled, I would have had my queen zoom back to safety in a flash. There wouldn't be a way to attack it.
I"m probably using the term "fried liver" incorrectly. I'm referring to something like in the Italian Game, Knight Attack opening or some variations of Petrov's defense where the bishop supports the knight in a K-R or Q-R fork early on... Something like this.
(EDIT: Oops. Fried liver is related to this, but involves the knight being sacrificed. Dang. All of those games I could have won if only I had responded differently to this scenario...)
This and wayward queen (which I don't like) are pretty much the only named openings I know how to use and defend against. If the opponent counters or does an incompatible moves... Well, something like this game happens.

Yeah Qg4 is probably not the best move at this stage...
Would you say that developing your pieces is more important than trying to use an early offensive opening? (Sorry if I'm bungling the terminology. I've sometimes struggle with the terms and notation.)

Got it. Thanks! Developing pieces > trying to get "gotcha" moves.
To think that all this time I feared the Scandinavian defense precisely because it brought out the queen early...
On a side note, I broke 800 for the first time since beginning!

Thank you so much for this analysis JayeshSinhaChess! This is immensely valuable for me, particularly the constructive criticism. (It doesn't hurt, because I know I need it.) I am going to really try and make the best use of this analysis.

To analyze your game.
1e4e5 2Nf3Nf6. This is fine, a relatively common and solid opening. 3.Qe7, the correct move from you, well done. 4.Nf3Qxe4+. Once again this is good play by you capitalising on your opponents overly defensive move. 5.Qg4. This is a poor move. Instead of focusing on development you began an easily defensible attack. You noted that you didn't want white to castle so you should have considered alternate options. 5e5, 5.Nc4, and 5.Bc5 were all significantly better as they developed your pieces and applied pressure while preparing to castle. 6.Rg1. Awkward move from white it defends against your threat but in a clunky way. Castling was a better option. 6.Bc5. Good move! you apply pressure, continue your development, and prepare to castle. 7h3Qe6. This is fine but forced. You've effectively wasted a move by being forced to retreat your queen. 8d4. Good move from white, your response was not ideal. Bd6 cramps your pieces and prevents you from playing d5 later on. If white played well here they could probably gain space and dominate the centre through a combination like 9.c4c6 10. Nc3. A better move was Bb6. 10.Rh1c5. Weak move from white which wastes time instead of developing pieces. Your response was solid although I personally would have castled before playing this advance. At this stage you should have been aware of the potential of hanging the now lightly defended and very in the way bishop on d6. 11Nh4Cxd4. Poor move from white again. It brings to mind the old but true saying that a knight on the rim is dim. The move that you were worried about (Ng5 would also have been poor as you could have easily defended it with Qe7. Also, you seem to be hoping that your opponent doesn't play in a certain way a bit too often. Try to account for all possibilities and then move accordingly. 12.Qxd4Nc6. This is a shaky move by white which you responded well to. Good solid development capitalising on poor positional play by your opponent. 13.Qd2Ne4. This move is fine but not great. Unless your opponent blunders here this is sort of pointless. Castling was better. 14.Qd3. and your opponent errs. Qe3 which would have pinned the knight and potentially allowed a queen trade was much better. Your response was a blunder. 14Nxf2Qf6+ can be blocked by 15.Nf3 which leaves white in a stronger position and ahead on material. Still as white blundered as well you gained an advantage. 16.Kf1. Not a great move from white as it leaves their king very exposed. 16. Nb4 is weak as it hangs the bishop. Furthermore just attacking pieces while they can easily retreat and consolidate is pointless and can easily allow for your to gain an advantage (usually in development and position).16bC5 was a much stronger move. Also note that if your bishop was on the a7g2 diagonal as it probably should have been you wouldn't have to make this move and would have a more solid position. This is why development is important. Even if pieces aren't initially involved in an attack you need to get them into positions where they can be when pieces are traded and the game becomes more open. 17Qe3+Qe7. Good response by you, this queen move while appearing defensive makes your position much more compact. 18. Qc3 0-0. Yay, you castled! Immediately your position looks much better. With your king safe and your rook in the game you have a huge advantage over white who, it should be noted, made yet another wasted move. 19a3Nd5. Fine move from white. As for your response I would have ignored the threat to the knight and played Re8 threatening to take the bishop on e2. While this was blockable for white it would have made their position very awkward. The threat to your knight could then have been averted with Nc6, a much stronger move than your choice Nd5 which leaves your knight unprotected and allows white to gain a tempo. 20.Qc4Ne3+. What is there to say here? awful move by white, its good that you spotted the fork. Note that if white had castled earlier this attack would have been a non-issue. Getting your king to safety early on is almost always the best option.
I'm going to stop analysing the game at this point as, short of hanging your queen it is completely won.
So what I would say in conclusion is that:
-It is a positive that you make plans however they should rely less on your opponent and more on you.
- Your tactical play is pretty good for a beginner. Spotting the fork and playing Nxf2 at move 13 (even though it was a mistake) were both decent ideas. It shows that you're thinking about what you're doing.
-You took far too long to castle. A stronger opponent could have gained a decisive advantage by attacking your vulnerable king.
- Both you and your opponent were too focused on creating immediate threats which, once again, can be exploited by more skilled opponents.
Please don't be discouraged by this analysis, a lot of your ideas in the game were solid and if you focus on mastering the basics and some early principles of chess like development and piece placement/strategy you will improve pretty quickly.

Follow-up: sorry to revive this thread, but I wanted to provide an update. Thanks to everyone's advice, my score has rocketed in the past week from the 700s to the high 800s-low 900s! I previously thought I wouldn't hit this goal for another year, if ever. A key change now is that I'm focusing just a *smidge* more on development instead of instanti "gotcha" openings, and it helps. I'm still *very* much a beginner with a long ways to go, but I feel like I've broken through whatever barrier was previously preventing my advancement
Again, thank you everyone for your help making this happen! ^_^
This was a long, drawn out game I just won where the engine says there were a lot of blunders and mistakes on both sides. The first half was a back and forth until my opponent blundered his queen, and the rest of the game consisted of me trying to convert my advantage into a win.
I'm not exactly the best chess player out there, but even after looking over the analysis, I'm still struggling to get why many of the bad moves were, well, bad. Can someone look at this game and tell me what mistakes we made, and why they were mistakes? This is a bit of a recurring barrier for to my improvement.
I greatly appreciate your attention to this, considering the length of the game.
Edit: annotations added.