On a budget, so:
Can I run a 9 speed front mech on 10 speed shifters?
Does a 9 speed chainset work, with 10 speed rings on, running 10 speed at the back?

Ta for experienced comments

Me I am just tight, but watching this thread

Yes you can use a 9 spd front end (rings and mech). But the chain, cassette and rear mech need to be 10spd.

Yes, and 10 speed front shifter works fine with old 9 speed front mechs (though you should obviously go 1×10 🙂

10 speed rear shifter and 9 speed rear mech however will definitely not work.

Edit: I can only vouch for SRAM stuff, I think Shimano 10 speed is a bit fussier.

I’m running a Sram 9 speed rings with an otherwise 10 speed set up, works perfectly. Very happy.