First of all, for any IKEA fans out there, this should not be taken as a strike against IKEA. Many different brands of flat pack/assemble it yourself furniture have the same issues.
No, glue won't get the dowels out :-) But not using glue can help a lot. Many assemble it yourself pieces of furniture recommend gluing the dowels in place and some even include a little bottle of glue. The glue is usually not necessary for basic functionality (cams, screws, etc. do that) but does help make the furniture a little bit more stable and secure. The glue doesn't just lie between the surfaces (as some adhesives on some surfaces will do) because wood is porous and the glue soaks in (I am sure there is a technical term for that), and I think the dowels, which are often tight-fitting to begin with (by design) may even swell just a little bit from the water in the glue. Which is all great for keeping things together but makes it really hard to cleanly take things apart.
I would probably try, a little bit, to get the dowels out. But if they break, cut them off with the saw of your choice (hacksaw, reciprocating saw, etc.) That may leave 1/2 a dowel in a hole, but that's OK because these dowels are not critical in and of themselves.
If you really want to get them out, you can use a drill. But be careful because a lot of times the dowels are in holes that only go partway through a piece of wood and have a finished surface on the other side. Better to leave them in there (but chopped off so they don't get in the way) than to take them out and mess up everything else in the process.
No, not to take the dowels out. To replace them. Once you reassemble the furniture, but skipping the dowels where they were not totally removed (and therefore you can't install new dowels properly), if there are places that are a little more wobbly than you like, a few screws in the right places - which will likely be near the old dowel locations - can do wonders for holding everything together.
As with removal issues, finished surfaces may be a concern. Back, bottom probably fine. If the dowels are into what becomes a desktop surface, screws won't look so god. In that case, I would typically use some small brackets with short wood screws on the opposite, hidden, side to hold everything together.