One way to find your internship is through the list of proposals. Another one is by discussing with your tutor or with the lecturers of classes you are most interested in. You may ask the latter what research exactly looks like in the course's area, for instance, in the form of representative open questions or main challenges. Both your tutor and the lecturers can guide you to researchers to contact.
Be proactive in looking for the internship. The time and energy you invest are definitely well spent. You should set yourself to have your internship validated (by your tutor) by the end of November. This way, the internship will start in due time in February, and you won't be overwhelmed with the homeworks and exams of the semester's second wave.
Your report should be clear and accessible. Do not forget that it will also be read by a non-expert. It should be easy to identify your contributions. Something specific you can do to increase the quality of your report is ask your advisor for feedback on a preliminary draft.
In general, scientific writing is a non-trivial task, at which you will gradually get better. There are ways you can speed the process up. Reading (introductions of) well-written papers on the topic of your internship should help; ask your advisor for those.
You can read the short essay How to write mathematics by Paul Halmos. I particularly like the following excerpt, which, to me, summarizes the most important point.
The purpose of using good mathematical language is, of course, to make the understanding of the subject easy for the reader, and perhaps even pleasant. The style should be good not in the sense of flashy brilliance, but good in the sense of perfect unobtrusiveness. The purpose is to smooth the reader’s way, to anticipate [their] difficulties and to forestall them.
Practice, by giving a talk at your team seminar, in front of your advisor, or with your classmates. In addition to the feedback you will then get, you can read (with a grain of salt) the first two sections of this guide by Ian Parberry (the third section is obsolete). Rehearse several times on your own, making sure that the timing of your talk is adequate.