Most cars have hydraulic valve lifters, that use engine oil to keep them adjusted. If one of these lifters is worn & the oil drains-out when the engine isn't running, it will click for a couple of seconds until it gets pumped-up with oil again.

Probably gaps in the valve gear. Expansion through heat closes them and/or if they are hydraulic the engine oil has to fill small cups. If it's quiet when hot that's fine.

Indicates poor maintenence, too infrequent oil & filter changes. Add a pint of marvel mystery oil to the crankcase and drive it 1K miles, then do an oil and filter change, add another pint of MM oil if you like . . .

not sure but make sure you have enough oil