Hi,
I would suggest you do some research on the different arrondisements. Many have the type of things you are looking for. With hundreds of hotels that will meet your other criteria, it is probably best to find where you want to stay first.
Also, I'm not sure of the rest of your trip, but do you definitely need the car? Since you will just be "parking" it, you'll be paying (rental fees) for something you won't be using. Not sure if you have any flexibility there, but if so...
Many hotels now offer internet connections, so I think you'll be ok there.
No need to stay in a hostel to meet other travelers Bars, cafes, etc. work great for that.
You might also want to give Priceline a try. Since you are open to different hotels, you can probably get a great rate there. FWIW, we are going to Paris at the end of this month and paid $115 for Novotel Les Halles.
Lastly, on our first trip to Paris we were about your age. We simply showed up in the city and found a hotel in a location we liked. We paid $50/night and had a view of the Eiffel Tower (the room was about the size of a closet, tho ). But, the point is, you can get a good deal that way and also see your room before committing to anything.
Have fun!