Skullbriar will get one +1/+1 counter each time it deals combat damage to a player (including you if its combat damage gets redirected), regardless of how much damage is dealt.

Skullbriar retains all counters, not just +1/+1 counters.

The counters that remain on Skullbriar as it changes zones aren't "placed" on Skullbriar. Effects like Doubling Season's and Melira, Sylvok Outcast's won't affect those counters.

Counters that adjust power and/or toughness affect Skullbriar's power and/or toughness in zones other than the battlefield. For example, a Skullbriar in the command zone with a +1/+1 counter on it will be 2/2.

Effects that last "for as long as that creature has a [kind of] counter on it," such as Aven Mimeomancer's, stop applying to Skullbriar once it leaves the battlefield. Even though Skullbriar retains the counters, it becomes a new object with no relation to its last existence in its previous zone.

Skullbriar's last ability only works if it has that ability in the zone it's moving from. For example, with Yixlid Jailer ("Cards in graveyards lose all abilities") on the battlefield, a Skullbriar with a counter on it in a graveyard loses that counter when it's put onto the battlefield. Conversely, that Skullbriar moving from the graveyard to the battlefield would retain that counter if Humility ("All creatures lose all abilities and are 1/1") were on the battlefield; if Skullbriar then left the battlefield with Humility still on the battlefield, it would lose the counter.

If a card becomes a copy of Skullbriar, counters will remain on that card when it leaves the battlefield (unless it goes to your hand or library). Once it does so, it stops being a copy of Skullbriar, so those counters will cease to exist when that card next changes zones.