tl;dr you may be actively harming yourself through your actions. It's a real roll of the dice as to whether or not it'll work out in your favour, but you can expect these qualities to be abused.
Relative to my new-hire peers, I feel like I tasked to do a lot more and I have to stay longer hours than most. I really enjoy the technical assignments that I am tasked with and I sometimes go above and beyond (staying really late to solve problems) at my own expense.
One might think that staying late to solve problems/tasks looks great, but flip it around without drawing on the knowledge you have in your head. You have to stay late to solve the work that your peers manage to get done during normal work hours. You lack a sense of boundaries with work, and a sense of priorities with life. In short, you're inefficient and don't know how to prioritize your life.
I do feel very jealous that my peers can leave work early and do relatively nothing all day while I am tasked with these never-ending responsibilities
Early or on time? Also, I would strongly avoid assessing your peer's performance if they are outside of your direct dependencies.
I don't feel like I should bring this up to management
Good, that would be a Career Limiting Move.
because it's generally not my peer's fault that they have nothing assigned to them, it is their managers
It's also none of your business.
My hope is that by being assigned difficult tasks and responsibilities, I will grow quicker career-wise.. but I feel like no matter how much I work, it won't be realized at my company
Maybe they are difficult, maybe they aren't. You're only a year in to your career, and all you know is that those tasks are difficult for you. You don't really mention your responsibilities, so can't comment there. No matter what though, I reckon there is room to take a step back and chill out. You've been there for a year and are already complaining about the lack of recognition for your genius.
I feel like there's more political games to play with management and upper management in order to get noticed for a promotion.
Hard work only pays off when the right people notice it. Also, a million good deeds worth of good will can be erased by a single misdeed.
I have a peer in this new-grad program who probably doesn't work 40 hours a week on their technical work (or even 40 hours, in general).
Not your problem.
Our manager seems to tolerate it however I am very jealous that this person can get away with this. In addition, this person networks a lot and disregards their immediate technical tasks. I am, again, jealous because I want to be able to network too without hindering my technical performance. I feel like this person will grow their career faster by, ironically, ignoring their technical tasks.
They might. You might. What they do is really not your problem.
So, I am not a very good networker and it would take me quite a bit of time to learning how to do so. I feel like I could make a bigger impact on technical work because I try to make strategic steps towards working for or being a consultant.
It sound like the above person's strategy is really not for you.
Answering your questions directly:
- Is my jealousy towards my peers, in general, valid? If yes, what should I tell myself to not let it get to me. If no, why not?
Sure, feelings are valid. They're doing things that are better for them, and you seem to be actively harming yourself and focusing on other people's work and not your own. As for what to do about it, that's really hard. Jealous is very difficult to manage. The most immediate strategy is to focus on what you need to do to be successful, and stop focusing on your coworker's (apparent) success.
- Is my jealousy towards the bonus peer valid, or again, should I just focus on myself? If yes, what steps should I make to have more time to network. If no, again, why not?
Same as above. It's one hundred percent not your business. If you want to network, learn to talk to people, ask questions, and be curious about them without sounding like you're interrogating them.
- Will working harder technically pay off, at all, or should I focus on making more management-related strategies to further my career?
I mean, technically yes, if you're continuously going deeper while broadening your experience. Get "comb shaped" skills. But really? Chances are high you'll hit a wall and burn out. Work normal hours, find a hobby, make friends, forge relationships, chat with coworkers on safe topics. When management sees you as a productive, competent, and whole person, you've got the best chances for success.