I have had more hot starts with JT9's during cold weather ops than with summer ambient temps. In that engine, at least, when you turn the fuel on (idle/rich) is dependent on EGT. Max motoring may go beyond just an N2 RPM if the EGT is above 100dC. The only influence reported temperature has is whether engine anti-ice is required after start.
Tailwinds, in my experience, didn't seem to have a big effect, if any, on JT9 starting temps, which was good with engine pneumatic air required for reverser deployment preventing reversers use during start. Hydraulic reversers are desired, at least for the placebo effect of blocking the tailwind during start and a quieter preflight experience (not only quiet, its also easier to preflight the fan blades when they aren't rotating at "max motor").
The Herc, as I recall, was more sensitive to strong tailwind starts. Lowering flaps to 100% in order to reduce wind up the tailpipe helped to control starting temps. A speed sense switch controlled the start sequence of fuel and ignition, so there wasn't much else the crew could do other than to release the start switch at the recommended RPM and hope for the best that everything worked as advertised.