You need different adhesive for those two floors. Work with them as separate floors. And you'll need an expansion strip where the two floors meet as both of those floors will expand and retract at different rates, so if you tile over them both with 1 tile (so the join is under the one tile) it'll crack right where the two floors meet.
Where do you need tile trim on a floor? Do you mean the door strip where it will meet another floor type? Or are you tiling a step or something on the actual floor? In that case a trim might be needed.
Also, what are those quarry tiles fixed to? For such a small area it might be worth pulling them all up and then sorting out either a floating floor to tile to, or perhaps fix tile backer boards all over the lot. Not too sure on what the practice would be there mind, bit above my pay grade (I'm not a tiler, I just make the tea for the lads
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Got any pictures of the actual floor for us?
Where are you getting your tiles from? That will limit / stipulate your adhesive options. No point us saying one brand and you can only get another, sort of thing.
