Adric, Mathematical Genius

If an ability is linked to a second ability, copies of that first ability are also linked to that second ability. If the second ability refers to "the exiled card," it refers to all cards exiled by the first ability and the copy. For example, if Fiend Hunter's enters-the-battlefield ability is copied and two creatures are exiled, they both return when Fiend Hunter leaves the battlefield.

Although Doctor's companion is a new variant of the partner ability, the rules for partner have not otherwise changed. Notably, Time Lord Doctors and cards with Doctor's companion do not interact with cards which have another partner ability.

The Doctor's companion ability allows you to have two commanders if one has the ability and the other is a legendary creature that is a Time Lord Doctor and has no other creature types. Creatures with the changeling ability, for example, can't be a second commander this way.

The copy will have the same targets as the ability it's copying unless you choose new ones. You may change any number of the targets, including all of them or none of them. If, for one of the targets, you can't choose a new legal target, then it remains unchanged (even if the current target is illegal).

Adric, Mathematical Genius's first ability targets an activated or triggered ability that is on the stack and creates another instance of that ability on the stack. It doesn't cause any object to gain an ability.

The copy is created on the stack, so it's not "activated." Creating the copy won't cause abilities that trigger when a player activates an ability to trigger. Abilities that say that a triggered ability triggers additional times won't apply to copying a triggered ability.