Can I run an Ultegra 10 speed triple chainset with a 9 speed chain and front mech?

Yes.

I have on my road bike:
105 rear shifter 10sp
Ultegra cassette 10sp
SRAM 1070 chain 10sp
Tiagra front shifter Triple from 9sp group
Sora front mech Triple from 9sp group
105 8/9/10sp rear hub
XTR Shadow rear mech 9sp
Tune Bigfoot cranks double
Sora little ring 8sp
Stronglite big ring 9sp
XTR UN91 BB

All works fine so can’t see you having any problems

Yes, might wear a bit faster but I’ve used that setup fine. Don’t worry about the mech but might be worth getting a matched chain and cassette if you can.

Its a hotchpotch for my touring bike.

10 speed Ultegra Chainset
9 speed Ultegra front mech
9 speed sram chain
XT 9 speed cassette
XT Shadow rear mech