Would I be editing the code of the plugin?
We wouldn't advise editing the plugin code as that would get overwritten during each update. Many themes provide a spot for custom CSS or the use of a child theme can be helpful.
Your main themes style.css file can also be used, but would be the last option as it, too, would be overwritten during a theme upgrade.
Wherever you decide to do this, you'll also simply include styling for the buy now and checkout now buttons as well.
Would I be able to create a dropdown menu that allows people to choose the size of the shirt they want to buy?
Product variation is available in the admin for each product. A dropdown display on the front end is not something we provide for in MarketPres (yet). You'd have to have something custom developed for that. Someone familiar with CSS should be able to assist you with that fairly easily.
Also, where can I delete the "have a user account" option? I think that somewhat confuses people since we don't have those.
Within the plugin we make the signup portion optional, but we don't provide the ability to completely disable the signup/login form in the plugin options. For now, this would have to be custom coded either as an add-on to the plugin or blocked from displaying on the front end.