How often does the mobile phone in idle mode detect the network?
While registered to the network or unregistered?
As per DRX it will scan for PCH, in case in non home PLMN, it will scan for HPPLMN as per SIM file for HPPLMN search timer ( probably 6 mins).
When UE stay without coverage, how often will UE do periodical searching of available Network?
Depending on Power control algorithm configured on BSC/RNC