A PR status does not expire or has any time limit to it. However, if a PR chooses to leave the country without a valid REP, the REP expires when out of the country. The PR status will be revoked.
As with any citizen in Singapore, as a Singapore PR, you are required to oblige by the Singapore law through paying of tax and contribution towards your Central Provident Fund (CPF).
Yes, as a PR, you can still stay in Singapore even without a valid re-entry permit. However, if you are not within Singapore, you will lose your PR status. It would be best if you kept in mind to apply for renewal of REP two months before the expiry of your current REP.
The general fee payable for REP application is S$10/year.