This turned out to be an interesting issue, so here's the TL;DR version of what I'm finding out.
I followed calderonr on points 1 to 3.
On point 4, I see the same thing – the upload/download buttons are greyed out for the intro puzzles, regardless of whether you are playing online. This is different than the regular puzzles, where the upload and download buttons are available online.
On point 5, it makes sense that the list of completed puzzles is different online versus offline.
When you're online, the Foldit server knows which intro puzzles you're completed. The Foldit server identifies you by your numeric user id. (You'll see you user number as the last part of the URL of your user page on the Foldit website.)
When you're offline, you're special user "00000000", and nothing gets stored on the Foldit server. The list of completed intro puzzles for User Zero is still stored on your computer however.
On point 6, I'm not getting the same results. I'm able to save the solution to an intro puzzle offline, then open the same named solution online. This is the same way that playing offline normally works. You do have to be in same puzzle to see a shared solution, and this applies to both regular puzzles and the intro puzzles.
The only difference I can think of is that my user id has completed all the intro puzzles, where calderonr's user id has not.
Even when I load a saved intro puzzle solution, it doesn't automatically complete the level. For example, Swing It Around requires 7700 points to complete. Offline, I saved a completed solution at 7736. When I open this solution online, it shows the same score, but the puzzle doesn't complete until I move a sidechain. (Simply pulling on the middle sidechain is enough, even though it doesn't actually move.) The puzzle completes after a move, even when the score is slightly lower than the saved score.
That's another reason why it may be easiest to just complete the intro puzzles again from scratch while online.
Opening an offline solution online is one form of evolver play. See Evolver on the wiki for how evolver play works. Usually, evolvers are online users who work on each other's shared solutions to the regular puzzles. Foldit also allows this special case, where an online user can work to evolve a solution shared by offline User Zero.
Evolving an intro puzzle is a really special case, one I haven't seen before. In theory, one Foldit instance could be shared by multiple players, a whole class, for example. The class could cheat on the intro puzzles once one student solved them. (Seems pointless, but….) Maybe the behavior calderonr is seeing in point 6 is intended to prevent that type of cheating.
That's the really too long version, but I'll keep looking at this as time permits.