1/ For reasons explained in other comments, I would search for a full-time job rather than an internship. To find small companies, check out the most prominent VCs present in the UK and check out what startups they have invested in (it's almost always publicly listed on their websites).

2/ No. Get a full-time job. While you're job hunting, work on a personal project that will further your skills and expertise. You will almost certainly be quizzed on that (at least, if you try to join successful startups)

3/ See above. In case you're interested: we're based in London, UK, and are currently hiring for 3 positions: https://pusher.com/jobs

Good luck!