I've had a summer internship after every school year I've had in college.
First to answer your questions, put whatever you want. If you're applying to jobs not related to your degree, well, it's up to you. Cater your resume don't have a one size fits all resume. Is it worth your time though? What do you ultimately want to do?
Yes, some early stage startups hire interns. It's normally easier to get a job at a large company offering internships. That way, you get the corporate experience early and realize how much better it is to work for startups.
If you apply out of city, consider costs maybe. Generally, it's more fun and adventurous and you're isolated so you end up learning a lot more somehow.
Negotiate payments when they tell you how much you're offered. Generally, if it's your 1st internship, you have no leverage to negotiate.
Failed attempts should just be considered incomplete projects. Don't list them as experience.<p>Now, my personal advice is this: if you know what you enjoy doing, try and be a little picky. If you settle for any internship just for the experience, you may become experienced in something you don't enjoy doing. Then, it would be rather difficult to express in your resume that though you are vastly experienced in one thing you really want to be doing something else.<p>Hope it helps.