In your original post, you state, " I have pulled a few pieces off to reveal the original unsanded stain and varnish." So, you have removed a few pieces of trim, is that right? Look on the back side and see what "color" the wood is. Can you determine if the front face is stained or not? Either way, stained or not, this determination would tell you, or help tell you, what stripping approach you would/you might need to take.
If paint is flaking or peeling off in some places, then maybe you can assume the paint layers have not adhered, properly, to the underlying finish coat(s). This would facilitate the paint removal. Your pics don't give any further details or info, as to exactly what other advice can be given to you, to further or better facilitate the paint removal.
If you have lots of trim to work on, I highly suspect it would be more advisable to remove, at least, some of the trim (the easier ones), strip them, then reinstall them. If the paint removal becomes difficult, while still installed, then I suspect you would have lots more difficulties stripping/removing the paint, than if the trim is removed, stripped, then reinstalled. You could work on one or a few removed pieces, to see how effective this procedure would work out, before committing to the whole task. If you do decide to remove the trim, to strip, make sure you label each piece, so that you know where it belongs, for re-installation.
Seems the main issue to how best to remove the paint, without affecting the underlying stain. Again, it's hard to advise which method may be best.... or most convenient, manual labor-wise. Part of this determination might be whether to remove the trim, after all, as opposed to working on it while still installed. You will likely need to test a few paint removal techniques, to see which works best, then determine whether removing the paint while the trim is installed or to remove it, to remove the paint. Personally, I would likely removed, say, one doorway trim, strip and refinish it, then reinstall it, before moving on to the next section of trim. Also, work on one door at a time, before moving on to the next door.
For any stripping work, I suppose you know to remove any hardware, before doing any stripping work.
Do you have any experience removing trim work, without breaking it? Do you know what might be the best tools to use, to facilitate removal? Removal, without breakage, may also depend on how the pieces have been installed and where..... and relative to an adjacent piece of trim. It's usually not hard to remove, but often you have to be careful, take your time. The proper tools helps with the care to be taken. The removal and re-installation process might be easier and faster, than you think, as opposed to leaving the trim in place, for the whole of the refinishing/remodeling task, especially if you have multiple rooms to refinish/remodel.