Hi, Kara
Good questions, and I don't know the answers for sure because I've never
done it that way. Why not give it a try and report back what you observe?
As to staining, my guess is that the methylene blue won't be bound
tightly enough to the DNA to migrate with it through the substrate. The
purpose of the marker dye isn't to bind to the DNA but rather to move
independently through the agar substrate at about the same rate as the
small fragments you're trying to separate. When the marker dye nears the
far end of the substrate, the smaller DNA fragments should be near it,
with the larger fragments strung out behind and the largest still very
near the original wells.
Staining after you run the separation identifies the bands by staining
them in place, so you don't need to worry about the dye migrating
separately. But again, don't take my word for it. Run it both ways and
see for yourself.
As to running whole DNA, my guess is that you'll run into difficulty
trying to get it to move through the substrate at all, but that depends
a lot on the type (size) of the DNA molecules, the voltage you're using,
and many other factors.
Best regards.
Bob