How do the Kobolds remember which parts are trapped?
Basically, this answer is about weaving the Kobold's own marking system into the narrative. It does assume you draw your own maps and don't use Dungeon Tiles or anything.
Obtain 6 or so pretty looking symbols (they don't need to have meaning, but if they look Draconic it's bonus awesome) Mark every square of the map with one of them. In the narrative, explain that the Kobolds did just that; they covered all the floors with all sorts of markings. For each room, assign 2 symbols to mean "trap" and the others to do nothing. Vary the symbols per room.
This will clearly signal to the players "this room is trapped", but then the word Kobold is basically a synonym for "traps" anyway, so that's okay. It will keep your players attention strongly on the traps, they will try to figure it out by watching Kobold movement (which is good! traps exist in the narrative to be interacted with) and will mostly stumble right into a trap in room 2 before they realise that the Kobolds vary which symbols mean trap in every room.
You could even tie a meaning to each symbol (in Draconic) and explain the words to players who speak it, and then have the chosen symbols make sense per room. Since Kobolds employ deadly traps in their home, they need a system to keep down their own losses, and that has to be simply enough to teach to their kids while still being confusing enough that confound and/or kill enemies.
To tie it even stronger into the narrative, include a room that the Kobolds use for training their young (probably one with few or no Kobolds in it, since they know it's less hurtful) where the traps just drop a bucket of water on their heads or launch sticky paper at them.
And maybe in the newest room, the Kobolds haven't had time to place traps yet, but they did place the markings. Your players will be properly trained by this point to move very carefully, figure out the system, and probably be amused when they realise the Kobolds have trained them like Pavlov's dogs to avoid certain squares, even if they don't do anything. (If they are trained so well they never trigger any, make sure they find a scrap of paper later that has instructions on which traps should go where that makes it clear none of them have been installed yet. Remember: if you don't tell them, it doesn't exist.)
(Regarding the last part, I once trapped my players in a room inside the Temple of the God of Theater, by giving them a huge grid full of letters whose only effect was that certain letters made the floating balls of electricity in the corners light up ominously. They were told that the room was "very dangerous and none had ever traversed it". It took them an hour to figure out they were being played and they loved it.)
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.
Best Answer
As SevenSidedDie says, there are no rules for this, so you(r DM) needs to make a ruling one way or the other.
Personally, if I was the DM on the spot, I'd probably rule that:
Yes, the characters can totally call out "invisible goblin on my 2 o'clock, distance 30 feet!" and be understood. If necessary, they can also use their finger to point.
No, I'm not going to quibble about whether the characters would know such tactics, or whether they'd really be appropriate for the setting, at least not without a very good reason. The characters, at least past level 1 in any typical campaign, are all highly experienced combat veterans; they'll know how to do that, or something equivalent.
Yes, relaying the information to the other players in squares is a perfectly reasonable abstraction. Or we could just skip that bit and just assume that the other characters now know where the enemy is.
Yes, the other characters can attack the indicated square. However, the target creature is still invisible, and thus has total concealment, so even by RAW, they'll have a 50% chance of missing it completely.
If the target is more than, say, 10 squares away, I might even give the other characters an additional penalty to hit, because they're not even sure exactly which square the target is in. Also, unless they're using telepathy or something, the target might be able to hear the shout and take evasive action, which might merit a further penalty to hit. Neither of those is RAW (as far as I know), but I'd consider them quite reasonable rulings, depending on the exact circumstances.
If anybody objected, I would discuss these rulings (especially the last one) with my players, and adjust them accordingly. However, I would also remind them that, whatever they can do, their enemies can do as well. If their wizard can pinpoint an invisible enemy's position exactly from 500 feet away, and relay it to the rest of the party, then so, presumably, can any intelligent enemy with See Invisible. That should, hopefully, be sufficient incentive to reach a decision that's actually more or less balanced.