I would be looking into the entire wiring loom and checking what works and doesnt while I am at it if I were you, (not as scary as it sounds).
The hazards not working could be something simple like the relay/ flasher unit being duff (very common) or just some wonky wiring, but as that and the battery/charging system are suspect there may be other issues and I think rust and wiring are the two main things you want perfect on a mini, as they cause the most issues.
But dont despair, theres a huge amount of knowledge on here along with people willing to help :)
I wouldn't - flasher will be down to low voltage.
If the wiring all looks nice and tidy and not butchered around like some of the horror stories we see, then I'd leave well alone unless you have experience dealing with automotive wiring.
Fine if you know what you are doing then there can be know harm in pulling apart and cleaning all terminations and earths.
Not necessarily down to low voltage, from personal experience the flasher units used are appalling quality, I have gone through about 3 or 4 in the last year and a half. As far as I can see there are no ill effects from swapping units to diagnose where the fault may lay.