How to best join baseboard to tile trim on wall

Sorry to say this, but number 3 really is your only option. First the reasons why:

1. I can tell just by looking and from your description, your subfloor is rotten and now unstable. More tiles and grout are going to crack, come up.
2. The mould is almost certainly inside the sub-floor and maybe even the jousts. Cleaning the surface will very likely not do much of anything except stir up mould spores into your house. Don't reuse stuff that has mould on it. Throw it away (including the tiles).

A few things to keep in mind: You cannot with 100% certainty identify what type of mould it is just by looking at it. It might be worth your time to have a sample tested. The mitigation methods for certain types of mould are much more stringent. However, you could just assume it's the worst case and mitigate it that way. In any case, you're going to want to have your house tested once all mould is removed and BEFORE putting new coverings or sub-floor back in. That's just one of the steps. Do some research on how to mitigate. It's not a simple process and it's not to be taken lightly.

Potential health problems from mould exposure can be much worse than many other things including asbestos. You'll want to do some research on how to mitigate this. Things like respirators are an absolute must regardless of the type of mould and full hazmat gear may be required recommended. You'll likely need to seal off the location and provide negative air flow to the outside. You'll likely need some air-scrubbing filters.

Sorry for the bad news, but mould is bad business. It's very likely already causing health issues that you may / may not be aware of as yet especially now that you've tried to surface-clean your sub-floor and stir up the spores. Regardless of whatever product you're using advertises, they don't always work. Do your research with this one, don't just buy a product / clean it and hope it's good to go.

Wikipedia

In architecture, a baseboard (also called skirting board, skirting, mopboard, floor molding, as well as base molding) is a (generally wooden) board covering the lowest part of an interior wall. Its purpose is to cover the joint between the wall surface and the floor.

https://en.wikipedia.org/wiki/Baseboard

The purpose of a baseboard is to easily cover the joint so that it does not have to be seen. It's much easier to finish the plaster and then tack on the board that take the time to ensure everything is flush and not get any plaster onto the board. I don't believe that they are "designed" to collect dust, that is simply a side-effect of using them.