1,2. There is no evidence that the room itself changed it's location. As it changes it's size and shape, the room probably exists in a somewhat separate dimension, similar to the Platform 9 3\4 as was already indicated or Moody's trunk with 7 compartments. Which means the room can keep it's location while open the exit to different spots every time. Thus, the tunnel does not move either.
3. We can't really tell, as we don't know all the enchantments that were cast to protect the castle and how exactly the protection works. Remember that during events in The Prisoner of Azkaban protection of the school was higher than usual as well, but the tunnel under the Whomping Willow was working fine.
We know, for example, that you cannot pass over the castle walls:
“I could climb a wall,” he suggested.
“No, you couldn’t,” said Tonks flatly. “Anti-intruder jinxes on all of them. Security’s been tightened a hundredfold this summer.”
But we know nothing about going underground.
4. There is a fair question about where the tunnel situated - is it outside the room? If we agree on the fact the the room itself is more of a separate dimension space, probably without any definite location in the castle at all, we can assume that the tunnel is just the part of the same dimension. Hence, we don't actually have the physical tunnel outside Hogwarts, it's more like a portal from one spot to another in a shape of the tunnel. It is just a guess, of course, as I don't think we have any canon explanation.
“It’s quite straightforward, really,’ said Neville modestly. ‘I’d been in here about a day and a half, and getting really hungry, and wishing I could get something to eat, and that’s when the passage to the Hog’s Head opened up. I went through it and met Aberforth"...
So basically it may be the same way the room opened exits to different floors in the castle, this time it just opened such exit in the Hog's Head.
5. I think - and again it's just my guess, as we don't have the canon source for it - the room reads inside one's mind. So if you ask for place to study DADA, you don't need to specify that you want cushions and textbooks there, it just provided you with the best option you could ask for. Hence, when Neville asked for food the room provided with the best option in his situation. Presumably Aberforth was safer option for Neville than kitchens possibly supervised by Carrows.