Yes, I do.
No, it's just saying that as part of exploration there might be some movement that is construed as travelling, but it never says that it's all the movement that there is.
He is not travelling, he is just moving around. A travel means something specific, usually long, and with a goal in mind.
Just read the DMG sentence.
It certainly is. Not many dungeons have levels 300 feet in size, that's 50 cases, most don't even go over 25.
And again, you can travel in some dungeons. For example back to the stairs. Otherwise, you are exploring, finding your way, sneaking around, fighing, discussing. Not travelling.
Yes, it's called "their travels" and in english it has a specific meaning.
The second sentence mostly talks about "overland journey", would you use that for a few hundred feet ? Would you say that adventurers need to catch their breath after walking that long ?
And yes, usually, when walking around, travelling, you can more a certain number of feet. But I don't think that you have ever applied travel pace while exploring a dungeon for the first time, and going from room to room, scouting, etc.