No. Unless you have some feat or special ability that allows you to take another action (be it a move action or standard action) during a move action, or an ability that let you open doors as part of another action, or allow you to do those actions as part of a full-round-action. Or a special rule that allows an exception to taking actions as another type of action, such as mount/dismount steeds as free actions with a DC 20 ride check, or channel energy as swift action as an attack action.

Otherwise, all move actions must start on square A and finish on square B, as long as your path isnt blocked in some way.

Alternatively, it seems that the intent of the game designers was that doors, unless they somehow block your path, shouldn't be that hard to open/close. We can read more about this on this post from 2009.

There is much discussion about "how long does one person takes to open or close a door while running" or "what kind of door is it?" or "what way does this door open". But, as a GM, you do have some backup on the rules if you decide that opening/closing doors should only take part of your movement instead of another move action.

Terrain and Obstacles

From tangled plants to broken stone, there are a number of terrain features that can affect your movement.

Difficult Terrain

Difficult terrain, such as heavy undergrowth, broken ground, or steep stairs, hampers movement. Each square of difficult terrain counts as 2 squares of movement. Each diagonal move into a difficult terrain square counts as 3 squares. You can't run or charge across difficult terrain.

If you occupy squares with different kinds of terrain, you can move only as fast as the most difficult terrain you occupy will allow.

Obstacles

Like difficult terrain, obstacles can hamper movement. If an obstacle hampers movement but doesn't completely block it, each obstructed square or obstacle between squares counts as 2 squares of movement. You must pay this cost to cross the obstacle, in addition to the cost to move into the square on the other side. If you don't have sufficient movement to cross the obstacle and move into the square on the other side, you can't cross it. Some obstacles may also require a skill check to cross.

On the other hand, some obstacles block movement entirely. A character can't move through a blocking obstacle.

You will see that many posters do rule it this way, opening a door takes 5 or 10 feet of movement instead of a move action. Personally, i would rule that if the door is open (ie: unlocked) it takes 10 feet of movement, if it's locked but you got the key, or if it's too big (like a double door), or too old (large dungeon metal or stone doors), it takes a move action.

If we're going with a strict reading, then if you're seeing the medusa's eyes from beyond its range, then you -- the remote viewer -- do not need to make a saving throw.

When a creature that can see the medusa's eyes starts its turn within 30 feet of the medusa, the medusa can force it to make a DC 14 Constitution saving throw if the medusa isn't incapacitated and can see the creature.

Your familiar will need to make a save, of course, if they are within 30ft of the medusa.

If you are within 30ft of the medusa and use your familiar to see, and you see the medusa's eyes, then it depends on the final clause of that rule: if the medusa isn't incapacitated and can see the creature.

Let's imagine you are a resource-depleted familiar-owner hiding from the medusa in an abandoned building. She is only 25ft away from you, but is unaware of your location and cannot see you due to being in total cover from a stack of crates. Then, you look through your familiar's eyes (a spider) to scout the area, but end up looking straight into her eyes.

In the above scenario, you do not make the saving throw, but your familiar does. It would potentially be a traumatizing event though, being in the mind of a creature as it is being petrified.

Otherwise, if you had failed your Stealth check and the medusa actually can see you, and then you scout via your familiar and end up looking into her eyes through the familiar, then yes you will need to make a saving throw.

Strictly speaking, as long as the four conditions are met, you have to make a saving throw.

1. can see the medusa's eyes
2. starts its turn within 30 feet of the medusa
3. the medusa isn't incapacitated
4. the medusa [...] can see the creature

It doesn't matter where your sight comes from as the first condition is not "that can see the medusa's eyes through its own eyes".

I'm reminded of spells such as Misty Step, which allow you to teleport to any unoccupied space you can see within 30ft of you (even if you're seeing through the eyes of your familiar).

As for the case of controlling another creature, that depends on how you are "controlling" them. If it is through Dominate Monster, for example, then only the creature you have controlled makes the save.

But more likely, you're asking about Magic Jar. If you are possessing the body of another creature in this way (your soul leaves your body and steals the body of another living creature), then from the wording of this particular spell, your soul and the possessed creature's body are considered as one creature. You -- the possessor -- must make the saving throw using the stats of the possessed body. If you fail the save, the body you are possessing becomes petrified. If that body also dies, your soul may leave that dead body and go elsewhere.

As for should your original body be petrified along with the possessed body, that depends on if your true body is within 30ft of the medusa. If it is, then you must also make a saving throw with your actual stats.