The counters that remain on Me as it changes zones aren't "put" on that card. Effects like Doubling Season's won't affect those counters.

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

Me's second 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 Me with a counter on it in a graveyard loses that counter when it's put onto the battlefield. Conversely, that Me 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 Me then left the battlefield with Humility still on the battlefield, it would lose the counter.

You must follow all normal timing rules when casting Me from your graveyard using Me's last ability.

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

Ashildr retains . . . I mean Me, the Immortal retains all counters, not just those granted by the first ability.

Counters that adjust power and/or toughness affect Me's power and/or toughness in zones other than the battlefield. For example, a Me in the command zone with a +1/+1 counter on it will be 4/4.