The Special Edition PHB disambiguates the wording. Some people refer to this as "stealth errata" since it doesn't seem to have made it into any other sources, nor was it conspicuously announced anywhere. Even the SRD doesn't seem to have been updated to contain the new wording. Nevertheless, it makes the intent of the feat clear. Emphasis mine:
When using the attack action with a melee weapon, you can split your move action
in that round in order to move both before and after the attack, provided that
your total distance moved is not greater than your speed. Moving in this way does not
provoke an attack of opportunity from the defender you attack, though it might provoke
attacks of opportunity from other creatures, if appropriate.
You can't use this feat if you are wearing heavy armor.
You must move at least 5 feet both before and after you make your attack
in order to utilize the benefits of Spring Attack.
Notably, this is still not the same as Pathfinder's implementation: instead of combining the attack and move into a single full-round action, it keeps them as two distinct actions but allows you to apply them in an unusual way (much like the Overrun writeup that you cite). But neither implementation allows you to make a spring attack using a single standard action.
If you want to houserule Spring Attack to take a single standard action, go right ahead. No one will stop you. I might consider such a thing myself in my game, if it ever comes up. But that's not RAW, and in one of those rare cases where RAI is demonstrable, it's not RAI.
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
Nothing says you cannot be mid-air when attacking. That rule simply doesn’t exist. So if you are mid-jump—i.e. mid-air—nothing says you can’t attack. So you can, provided you meet all the requirements that the rules do have for attacking (actions, armament, reach, and so on).
Note that you must have some way to attack mid-movement in order to actually attack mid-jump. Spring Attack covers that, for example.
Anyway, the issue is not addressed explicitly anywhere, which limits how much it can be truly backed up. But Tome of Battle’s Stone Dragon discipline says that “Unlike with other disciplines [...] Stone Dragon maneuvers can be initiated only if you are in contact with the ground,” (pg. 81), which suggests strongly that the default is the expectation that you can attack mid-air. The Tiger Claw discipline has many maneuvers that involve making a Jump check in order to gain some benefit to your attack, and are described as jump attacks (though mechanically you don’t actually move).