First, I'd ask you to be a little kinder to yourself. You are apparently working full-time and going to school, and you have been doing this for several years. While doing this, you are getting a handle on your money and making cuts that also cost you time and energy (e.g., biking to work, hunting for street parking). No, you are not there yet -- but you are headed in the right direction. So you should be proud of what you have done so far. Sure, you made some silly decisions with undergrad loans -- who hasn't? Really, are you the only person who ever made a stupid decision when you were 18 or 20? ;-) The key to success is not never making a mistake; it's making sure that you are always making new mistakes.
So on to the plan: you graduate in May. Is that degree going to qualify you for a raise, a promotion, a different/better/higher-paid job? I assume so, because why else would you be working so hard to get that degree? So think of this in phases. Phase I is now. Looks like you have about $500/mo. So first things first: get yourself an emergency fund. Set aside money for the next couple of months to handle any upcoming expenses: do you have another semester's tuition to pay (I am assuming you will take the upcoming $4500 and send it directly to next semester's tuition, but if you need to cover the tuition first, what's your plan)? What about car repairs, oil changes, getting a flat? If you will be looking for a better job, do you have the wardrobe you need? Etc. You need a little money floating around in a checking or savings account to cover life's variabilities.
Once you have your little slush fund, I actually agree with the advice to knock out the smallest loan* -- if you can get even one loan done by the time you graduate, you will free up not only your time, but also room in your budget. If you can get both of those two, even better. And personally, I would not worry about the car right now.** You have a lot going on, so focus on your work, your school, and managing your other spending/keeping to a frugal lifestyle in general.
Then you graduate. This is Phase II. I am assuming you plan to stay put with your company, because you usually have to commit to 2 years or so to get the tuition reimbursement. But that's fine. Now you are presumably in the job/career you have studied for and are basically entering "real life" post-school, so this is the time to plan your longer-term financial approach. So first, sign up for PSLF (if you haven't already). It may or may not be around, sure, and you may or may not decide to stick with it the full 10 years. But why not keep your options open, since it doesn't cost you anything?
Beyond that, go look at the Investment Order sticky -- it will tell you what to do. Generally, that means 401(k) up to the match, and then tackle those high-interest loans. In your case, I would also qualify the second part of that: when you focus on your high-interest loans, start with the private loans, because those don't qualify for PSLF. So why not start with the loans you know you're going to have to pay off, before worrying about ones that Uncle Sam may pay for you?
This is also the time to think about the car. If you do not need it for your long-term job, then by all means, ditch it. Even if you do need a car, you can probably do better selling it for what you can get and buying a little used econobox -- really, if you're street parking in the city, do you want a nice car anyway?***
Then, at some point, you will no longer be required to stick with your current company because of the tuition reimbursement. This is Phase III, and this is when you make the decision: do you want to stick with the PSLF path, because you like your job and you think the program is going to stick around? Or can you get a sufficiently larger salary elsewhere that it justifies dumping the program and just paying them off yourself? Note that we are probably 2-5 years down the road, though. This means that you don't actually have to worry about this right now! Sure, if you get the private loans paid off, go ahead and turn your focus to the high-interest public loans. But worry about whether to leave the program when you are actually presented with the option of a higher-paid job in the private sector.
So, in short, triage. You have a long life ahead of you, and a not-extravagant lifestyle (especially considering that more than 25% of your expenses is simply paying off the education that earned you your current income). The key to success for you just comes down to not inflating your lifestyle once you have more money -- keep focused on paying down the loans one way or the other, start saving in your available tax-sheltered retirement plans, and then just make sure that all of your future raises and bonuses go toward those same causes, and you will be more than fine.
Finally, the house: do not even think about this until Phase III. (A), yes, you have loans that are more than your current salary, and a car payment to boot. But also (B): at some point you may want to look for a different job, and so you do not want to tie yourself down to a specific location before you do so.**** IMO, you should not even consider buying a house before you (1) have a paid-off car (or no car); (2) have a long-term job; (3) have saved at least a 3-month emergency fund, and (4) either have fully committed to the PSLF path, or have paid off all of the loans above about 4% (which, btw, is only about half of your total loans and is totally doable in a year or two once the car is gone and school is done).
*Note that this is completely contrary to my standard advice and is only because we're focusing on what you can do in the next @9 mos.
**I know, I know, I am being so unlike me today!
***My boss used to take the train to DC every day and park on the street by Penn Station. He made several hundred thousands a year, and he still drove about a 15-year-old Corolla, because he didn't want to have to care if it got broken into or dinged.
****Ask me about my condo in downtown Baltimore that I bought immediately before they sued the builder and then couldn't sell and so rented at a loss while I moved for a job in DC and then out-of-state, before finally selling @5 years later for a 20% loss. A/K/A "The Albatross."