1) Does your vocabulary need to be excessively good?
It certainly wouldn't hurt; generally though you don't need to go out of your way to learn 'fancy' words. Instead of increasing your vocabulary in a general sense in hopes of finding words that might be relevant, you'll probably just pick up words that suit your text as you read it, and support material related to it. Really all you need to do is avoid colloquialisms (and idioms, if you can) because they tend to detract from the overall 'professional' vibe you want. 'Exuberant' is great, but hard to fit in comfortably, 'happy' is more broadly applicable but weak, 'pretty alright tbh fam' - probably not.
It's mostly about choosing a word or phrase that suits what you're writing as opposed to showing off how many words you know. Bonus points if you spell them right.
2) Do you have to write an essay each week to keep up your writing skills?
Nah probs not. Honestly I only wrote extra imediately after reading the text (to get a feel for what I would write about in the future) and in the time approaching SACs. I didn't write very much extra before the exam; partly because I was a bit starved for time, but also with literature you don't need to memorise anything - because all the material you need to do well on the exam is present on the exam, and you will have been studying your texts extensively over the year, the extra work then probably isn't worth it. That being said, if you feel like writing an essay every week, go ahead. I'd only caution you that you might tend to fall into patterns, or write the same things every time as a result; I found personally that writing too many practice pieces took the fun out of writing an original piece for the SAC. Then again, most people probably wouldn't find it fun to begin with, so it's really up to you. You certainly don't need to, imo, though. Just smash out a paragraph when you hear something interesting and/or a new idea that you haven't thought about and you'll be golden.
3) Do you need to read lots throughout the year (things which aren't related to your texts)?
I ceraintly did, but mostly for fun. Read my texts periodically throughout the year (+ texts for Classics) and essays, criticisms, other people's notes, context, etc., but otherwise I would just read what I felt like reading. If you don't like to read, or don't want to read in lieu of doing something else, don't worry. Given that you've picked literature, the extra reading probably won't really improve your skill that noticeably.
4) For those who have also studied Literature alongside English, is the amount of essays written overwhelming?
Didn't do English, but I did Classical Studies, which is another essay class with lots of extra reading. Sometimes it could be overwhelming; the last thing I wanted to do on my hand-earned weekend or scant afternoon free time was to write essays or even short practice paragraphs because it's so tedious. With non-essay classes, it's pretty easy to sit down and do a set number of questions, but in my experience, essays require a lot of extra thought and effort. At the start of the year I tried it anyway, but as things went on a bit I just settled for 'this time I'll do literature, next time I'll do classics'. It certainly didn't hurt the quality of my writing, and it was a lot less mind-numbingly boring/anxiety inducing/generally unpleasant than trying to keep up. Keeping in mind that Literature is less work than English towards the exam period (and you get some extra time), it shouldn't be overwhelming if you're confident with your writing ability. Probably worth waiting for someone who did do English/Lit for a more relevant response, though.