Quick and dirty answers.
Tappet clearance MUST be provided or the valves will almost certainly NOT close. This means gas leakage past the valve seats and very quickly burnt seats, yes even in cast iron burnt seats, and naturally low compression and hence low power output.
Hydraulic lifters are a means of providing this clearance but doing it quietly. Check the noise from the engine of one of the local hoons who has replaced the hydro lifters with solid lifters.
Oil pressure has next to nothing to do with the operation of hydro lifters. The lifters are ported to an oil gallery but the lifter port closes immediately the cam lobe commences lifting the cam follower. As long as there is a hint of oil pressure the hydro lifters work - the rest of the engine will likely go bang though!!