So, my questions are: 1) which goes first, the epoxy or BIN?
The BIN (I mean shellac generically here, I don't want to specifically recommend BIN as there are better ways to get your shellac*). The shellac is acting as a sealer/primer in this case, with the epoxy acting merely as filler.
However in theory you can seal/prime with epoxy, because like shellac it is a very good sealer (arguably superior to shellac in fact, assuming the correct application).
2) Can I leave the deeper veins as they are and just clean them out as best as I can?
Can't say, sorry. Sealed well it may not matter that some is left, but I believe good practice is that you should dig out all of it, or remove the offending pieces and replace them, unless this is impractical for one or more reasons.
3) When I dug out the sappy parts, the wood is pretty rough - will the epoxy better adhere to a rough surface or should I sand the gouges smooth?
Rough is fine, actually preferable. Although in this context I'm not sure if it would make enough difference for it to matter.
4) I know sanding will be required once all is said and done - should I retouch the Woodlife?
I think it will be called for, yes. You can in theory smooth off epoxy fills and leave them dead-flush, using for example a plastic squeegee or palette knife, but in practice I find you really have to leave epoxy slightly proud of the surface and then make it flush after it has set.
Note 1: epoxy goes 'hard' long before it has fully cured**, at which point it has become as hard as it can get. Usually it's best to work it when it is harder rather than softer, but if you work carefully you can pare away the bulk of the excess with a very sharp chisel when it is only just hardened (e.g. after approximately 1 hour with 5-minute epoxy). However, my usual method for flushing epoxy fills is to wait for a full cure and then plane down with a hand plane set for a very light cut, prior to sanding to make it completely flush and smooth.
Note 2: you may want to do a few comparative tests but you should find that filling the epoxy with some sanding dust (or purchased wood flour) will make it easier to work, which simultaneously lowering costs slightly.
*Shellac is much better bought as dry flakes or buttons and made up fresh in small batches than bought readymade in a tin.
**Even with fast-setting epoxies it's usually good to wait at least overnight prior to sanding, but waiting 24 hours or longer won't hurt.