"Do companies (especially large ones) even look at paper applications anymore?"
Yes, and double yes. Most of these large companies actually work in a franchising system where the local store manager is in charge of hiring, and most of them find looking through a few paper applicants quicker and easier than going online. It also assures them that the applicant can come in and isn't just blanket applying for everything, even jobs they can't show up for.
"Does showing up in person do anything for my chances?"
Very much so. I worked at a Subway and every time somebody would hand in an application, we would give input as to our first impressions. Dress in a clean shirt, smile, and act like you can take orders. Although remember to ask for the manager. Don't just give it to anybody, bets case it gets to the manager and they don't get that first impression thus forget about you. Worst case it never sees the managers hands.
"Am I just wasting time and gas doing these things?"
Most of the time, yea, you are. Managers find it easier to just pick up the first resume off the top of the pile, therefor the most recent and most likely to still need work, so unless you get lucky and they are hiring at the time your hand in may never count. However all places have to hire at some point, you just need to be vigilant and keep handing in.
Best trick is to ask for the manager and ask the manager if they are hiring. They may not be, but they might direct you to somebody who is. Also, remember to include your available hours on the resume.