Just my personal opinions, but here they are;
Just wondered what your FTO teaches with respect to use of landing lights for day operations?
Switch on at HP/pre-departure checks, switch off with after-landing checks.
What are the reason(s) for using landing lights by day?
Pretty much the same as DB6, if you are approx. in someone's 12 O'clock, you'll see the light easier than just the aircraft.
Are there any airfields where the use of landing lights in the circuit by day is mandatory by local diktat?
Not that I go to, but it's not a bad idea to do so. That doesn't mean I think we need more petty airfield rules.
If your FTO teaches use of landing lights by day what is the turnover rate on landing lights?
They generally need replaced before a 50hr check, that's on PA28s/38s, mounted in lower engine cowling, may last longer on wing mounted types.
What do you think of teaching landing lights off after take off at 300 ft aal and the switching them on again during the pre-landing checks?
I agree with CGB, pointless switch fiddling in the circuit isn't helpful, students have got enough do do already.