The triggered ability triggers both when Maulfist Revolutionary enters the battlefield and when it dies. You don't have to choose only one.

Maulfist Revolutionary's triggered ability gives only one counter of each kind. It doesn't double the number of each kind of counter. For example, if a creature has two +1/+1 counters and a charge counter on it, it gets one +1/+1 counter and one charge counter.

To give a counter is to put a counter on a permanent or to have a player get a counter. Effects that interact with a player getting counters or counters being placed on permanents interact with Maulfist Revolutionary's triggered ability.