-"Read & Write Mode" means you can resume recording on a movie by loading a savestate, instead of just watching it. Just watching is called "Read Only" mode, and if you load a savestate it will simply play the rest of the movie from this time, but not record anything new.
-I don't think there is an option to rewind frames, you can only go forward in PSX. That means if you want to make a new savestate from a few frames earlier, you instead have to play the movie from the start, and then make a new state when you get to the right bit. Afterwards, then resume recording from this new save state.
It is strongly recommended that you use way more than just 1 save state, otherwise it's highly unlikely that your TAS will have the standards required for this site. It's not hard to use more than 1 save state is it? Just save parts of your run to slots 2-9 too, use as many as you need to test each section. Using just 1 is bad, since it assumes that your current attempts will automatically be the best, and that you won't need to ever go back to an earlier section to redo something. Using more than one is far more efficient, and will make your TAS much better in the end. Basically, make a new save state each time you need to make a decision, or at anything which needs to be tested. You'll need to check to see if something actually works, and then reload to either accept that input or try another one; thus a save state is needed.
There's no need to use 6 different save states when navigating through menus for example, I'm sure 2-3 would easily be enough. 1 could be the very start of the menu, and 2-3 just the individual states you use for testing each frame. Once you're sure the menus have been done properly, you can then change state 1 to be the start of level 1 for example, and then use 2-9 for whatever you want.
But seriously, you won't get anywhere with just 1 save state, use more.
-No idea about this problem, but it sounds like you're not resuming record properly? But that shows what I mean when you make a 'bad save state'. If you had a problem with save state 7 or so, you could just reload from 6, or 5 for example. With only 1 state, you'd have to start the movie from the beginning, and watch 10 minutes just to get where you want. If you were using more than 1 state, you could play the movie, and then just load 1, or 3, or 6 etc, and then you would only have to watch 1 minute of your run instead. Clearly more save states are better, and you can get to the bad section a lot faster.
But just to make sure you're trying to resume record properly, this is what you should do. Play your movie (read only mode should be on by default). If you have the savestate you want to resume record from, then go into File->Movie->Read Only, (or press shift+8), and then load this savestate. This will resume recording from this point. If you do NOT have the savestate you want to record from, you need to create one. So play the movie (Read only), and then when the section is coming up, be prepared to make one. Trying to make a new savestate at the correct time while the movie is playing is kinda silly, so you'll need to slow it down to make sure. Try using pause/frame advance when the section is coming up, to make sure you get to the right frame, and then make a new save state. Then put Read Only mode off (do Shift+8, or click in menu) and load this new state, and you can resume recording from it.