If you are going to do this yourself there are a # of tools needed. If you only ever do one gear box in your life it will be hard to break even. If you see yourself learning a little more and doing at least two then you can break even or even come out ahead a little. The pros that rebuild these boxes also do a better job the first time. What I'm trying to say is it won't save you any money to rebuild this box yourself. But if it's want you want to do then there are people here who will try to help.
You need access to a press and the tooling common to those. Particularly a small size bearing splitter. I used a press once the guy not only didn't have any bearing splitters but he didn't know what they were. Don't know what he did on that press.
You need the output flange remover tool from Cycle Works and the measuring plate from them also and get a package of the assorted shims. With the ones in the bike now and the assortment you should have plenty. I also like to use the parallels in addition the the plate. And you will need a depth micrometer.
You may want to wait till you have it apart or just get all the bearings now. Ted Porter will sell you all of them and gaskets and seals and a few other things. But I don't know if we have the lower price yet on the shift kit in this country. It is being sold by MotoBins in England for less than I've seen it here. You can wait till you have it apart to see if you need the kit of course. Once you have it out it is recognizable by several parts that look different, particularly the pawl.
I don't know of anyway to tell for sure if your bike has the kit or not except to look at it. I did notice when I installed one the shift throws were longer but it's not something I measured or think anybody else has either. I think because you report there are false neutrals now it might be a good idea to assume you don't have the shift kit. The kit is supposed to help fix this problem, ain't it?
About the bearings; it is pretty standard to replace them all if you want a really fresh box but the front input shaft bearing is not always replaced. You can also take the path of just replacing the ones that feel ruff or loose and again the front input shaft bearing is usually left in place. It can be replaced but it is a proprietary and is only available from BMW. It is not a ball bearing but a needle bearing. I guess this is why it lasts longer.
Another thing about bearings. They are not the run of the mill generic bearings you might get from several sources. You will have to buy parts and bearings from suppliers that sell the right thing. All I remember about this is if we get into it the term "C-3" will be thrown around a lot. And some people will advise to only use "BMW" bearings. But if you stick to C-3 most will leave you alone.