1. It certainly doesn't hurt and adds to your case. It depends on what actual evidences you have of your relationship.
2. I would not apply for a partner visa from a bridging visa. Food for thought: you're on a BVA because you're waiting for a decision on the 2nd WHV. What do you think will happen to that BVA when you withdraw the application?
3. No. If you have de facto evidence for that entire 3 year period prior to applying, a CO may choose to grant the 801 immediately after the 820 and you skip the two year waiting period but they may consider your long distance in the beginning to be dating, not de facto. As far as relationship length putting you ahead in the queue? Nope.
4. You can study when you are on a visa that allows it. You'll pay international fees until you are a permanent resident and I believe even that is in the process of changing to domestic fees only for citizens from what I've heard.
5. If using an RMA sped up the process, I guarantee you everyone would use one. It will make no difference. Your case gets picked up when it gets picked up. It's luck of the draw as far as wait times go. Now, if you aren't willing to put the time and effort into the research to do the application properly, using an RMA would be advised. In your case, if you're trying to prove your relationship for more than 3 years to get the 820 and 801 at the same time, then you probably should so your case is put together well in your favor. Use an RMA. You don't need a lawyer. Though some will be both.
If you get the 2nd WHV, that doesn't mean you'll have to wait a year to apply for the partner visa. In fact, I believe you'll need to wait for the WHV now as you should not apply for the partner visa on a bridging visa (that comes with it's own difficulties that will require professional help). You can still apply for the 820/801 on the 2nd WHV. You'll be granted a BVA straight away but it won't go into effect until the WHV expires, so you'll be bound by the conditions of that WHV until you are no longer on it. Then the BVA will have full work rights. It is possible the 820 could be granted before the BVA even goes into effect.
Once you apply for the partner visa, you're eligible for Medicare.