Hey Bev —
Glad to hear that you are finding the thread useful — here are some thoughts and answers I hope you find helpful —
1) You are not given scratch paper, and, instead, you are allowed to write in the LSAT booklet. Not sure if you have any copies of official exams laying around, but what you see in the LSAC books is exactly what the actual test will look like. A fairly recent change is that each game is presented on two pages (as opposed to just one), leaving you plenty of space to diagram as you’d like.
2) In terms of timing, absolutely the most important things for now are developing your big picture sense of how games are designed / work and habitualizing effective diagramming methods. As you get deeper into your studies, as you stated, you will naturally get faster and faster at solving problems. But if you worry too much about quick ways to solve problems before you’ve had a chance to develop your big picture/diagramming abilities, I think you expose yourself to a lot of inefficiency and unnecessary trouble — it makes more sense to develop your fundamental big picture / diagramming skills first, then devote yourself to getting better and better at applying that to solving problems.
3) Having said that, of course it helps to have a strong sense of how games ought to go, how inference chains ought to work, and so on, and I think one of the very best ways to do that is to play the same games over and over again. I suggest you start with basic ordering and grouping games that you didn’t find particularly difficult — play them again and again on separate days, with the goal of doing so just a bit more efficiently and quickly. I think doing so will really help give your brain a visceral sense of how to order things along a line and make inferences about order, and how to place things into groups and make inferences about those groups — and I think that can help get your own juices flowing better in terms of creating/sensing the right types of instincts for how to approach and solve games.
Hope some of that helps and good luck — if you have any follow-up now or down the line just let me know —