I understand that for normal animals this is necessary, but what about the animals linked to the player? All the rules I see say Handle becomes a free action and Push becomes a move action, but it is never said if the player still has to roll to command their animal companion.
You do still have to roll. The important detail is that you get a +4 circumstance bonus to Handle Animal checks with your companion, and you only need to get a 10 (12 if the animal is wounded) to use tricks your animal knows.
That means you only need +7 in the ability between skill ranks and Charisma, and it becomes impossible to fail for tricks your animal knows (remember that a 1 is not an automatic failure on a skill check). For your animal companion, a handle is a free action, so it doesn't take any time from your character to do it.
For pushing it's a lot harder to make it a guaranteed success, but as you get bonus tricks for your animal companion it tends to come up pretty rarely in my experience. If you do have to do it, it's a move action for your companion, so you can do it and still cast a spell or attack something in a round.
Yes, you do. Since there's no rule saying you don't, then QED. I GM for a druid PC where we use that rule, and after a couple levels it doesn't come up much - it's basically "spend an action if you want the animal to do something totally freaky it wouldn't normally," which is fair.