If a permanent has protection from Rats, it means four things: 1) Damage from a source that’s a Rat that would be dealt to that permanent is prevented. 2) Auras and Equipment that are somehow Rats can’t be attached to that permanent. 3) Rats can’t block that permanent. 4) That permanent can’t be the target of abilities of Rats or spells that are somehow Rats.

Nothing other than the specified events is prevented or illegal. Hungry Lynx can’t block a Rat with the ability “This creature can’t be blocked,” for example.

The opponent who creates the Rat token is the owner of that token.

Tokens that are sacrificed or destroyed are put into their owner’s graveyard before ceasing to exist. If the token was a Rat, Hungry Lynx’s last ability will trigger.

Hungry Lynx’s last ability puts a +1/+1 counter on each Cat you control, including itself.

If Hungry Lynx dies at the same time as a Rat, its ability triggers, but Hungry Lynx won't be on the battlefield as that ability resolves. It can’t be saved by the +1/+1 counter that would have been put on it. The same is true of any other Cats you control that die at the same time. Your remaining Cats will each get a +1/+1 counter.