odd problems you have there.
the oil pump is a pain on a toyota 2C, and it does require removing the sump etc. If you do remove the sump only seal with proper toyota FIPG or threebond 1208 etc. Do not use general purpose RTV in rainbow colours.
if the oil was not travelling upto the head - immediately stop driving the car, you will completely destroy the engine.
For cold start device - clean it out and retest it or replace the whole item.