I don't think that you are nit-picking, in that I think your concerns are valid ones, but I do think you need to communicate more clearly and directly.
I get that 'just mentioning' things feels less 'naggy' than coming out and saying it, but at the same time I think it runs the risk of the other person not remermbering, or not realising thow importnat the issue was to you.
The heater issue you seem to have manged well - you agreed to her request, told her what the conditions were, and she has chosen to try a different solution, which is fine.
The tea-towel - I think you need to speak directly.
"I noticed that you'd mopped up using the tea towel - I appreciate you clearing up your spill and I realize that in an emergency it's natural to use the thing nearest, but I really don't like having tea-towels / clothes used on surfaces used on the floor. I'm sorry, I probably didn't explain that before. The mop / floor cloths are kept here . New washing up clothes are here." you can also take the opportunity to tell her any other routines you have (e.g. do clothes used for wiping kitchen surfaces become floor clothes when they are past their prime, or do you colour code and never mix the two? Do you want her to tell you / add a note to the shopping list on the fridge if she takes the last cloth out of the package? Are cleaning clothes for the bathroom stored there or in the kitchen, and do you have any rules / preferences about them?)
And consider asking your lodger for suggestions - she may have preferences too, and a little bit of give-and-take might make it easier for both of you- you'll feel less like you are nagging, and it gives her the chance to raise it f there are any things which are irritating her.