As far as I have read, Treasure Stitching just turns an object from 3D into 2D and doesn't move it into another dimensional space. That means a Bag of Holding could be put on to a cloth and the Treasure Stitching spell safely used on it. The cloth could be folded up and put into a normal bag or pocket.
However, what if that Treasure Stitched Bag of Holding was then put into a Portable hole?
The rules say putting such bag in another such bag or a bag in a hole is a bad idea and all sorts of nasty things happen. (You get sucked into the void and have to draw a new character....or is that Talisman?)
Does the spell prevent this from happening or does everything explode as normal?
"When you cast the spell the entire pile disappears into the cloth, replaced by a highly accurate, sewn picture of the pile from whatever angle you wish."
It's an extradimensional space.
Also, have a think about what the D means in 3D and 2D. There's dimensional shenanigans going on here any way you slice the cake. Bag of holding is not compatible with this spell.
*edit*
Actually, looking at the extradimensional space section example, even though they aren't compatible, that probably only causes the bag of holding to no longer be accessible when it is in the cloth. So, it may not really matter. Depends whether your GM rules the bag of holding overrides the treasure stitching or vice versa. I don't think the rules are clear enough to distinguish.
Portable hole shenanigans depends on whether 'deactivated' means all interactions or just the ability to remove items. The answer to your question is the same as the answer to this one:
If you bring a bag of holding and a portable hole into a rope trick, do they still cause the rift?
My answer would be no, they are inactive --- but the rules are silent as this is such a corner case.
If you take any container with an extra dimensional space into a dead magic zone, I think it slams shut till taken out. The outside is made inactive stopping access to the magical part.
What level is treasure stitching? I think 3rd level minimum to affect a magic item like that.
A while ago I posted my ideas about adventuring in 10D space. Everyone hated it. By rights, Phase Spiders should lair in 4th dimensional space. Do the contents of a bag turned 2D extend into the 4th, 5th, 6th, 7th, 8th, 9th or 10th dimension, or do they extend into the same extra dimensional space as the portable hole? The interaction of the portable hole and bags of holding are because they are interacting in extradimensional space. Either everything gets lost in a place that does not exist, or it all gets dumped back in the astral plane.