Built-In Shelves/Bookcases - IKEA vs Built
I have experience with both.  I'll try to answer your bullets and then some other thoughts.

Which adds more to a home's value?  Well, typically bookcases don't add to the value because some people want them and some don't.  If you wanted to build them in, you're typically going to go custom with that and build it yourself.  The rigid dimensions for a pre-made bookcase will normally not allow it to look truly built in.  Also the particle-board-covered-with-paper finish is not going to look good in the long run and could actually detract from the home value because of the cheapness.

However...  I have used the more modular Bestä system to make a built in TV entertainment area that looked very custom and fit the area well.  The reason this worked better is that the "lower quality" IKEA frames were mostly hidden by doors that have better materials and finishes.  I wouldn't have ever built in a Billy.

Which method is stronger?  This is going to largely come down to your building skills and materials, but custom can be stronger.  However, Billy bookcases when properly assembled will be strong enough for their purpose.  Any weakness will only show when it's time to move the bookcase.  They don't hold up to twisting and racking forces as well because they are not glued and only use a few locking fasteners.  They are great for sitting in the corner and holding things tho!

Which is easier to paint or stain? Custom is going to take the lead here for sure.  IKEA uses a particle board covered with a paper-based finish.  This can't be sanded or easily repaired if scratched, nicked or drilled.  It's designed for you to leave it the color it was made, and they do come in a variety of colors.  I'd worry about adhesion issues and what happens to the paper finish when it gets wet (if using water-based paints) and it can't be stained because it's not wood.  There are a few solid wood items that IKEA makes, so none of this applies to them.  They are just wood and you can do whatever you want.

So, you say specifically you want a built-in project.  Unless this is a very low cost home I wouldn't consider Billy bookcases.  They can't be easily painted and can't be easily repaired (like if a shelf peg breaks out).  They just are not intended to be that type of furniture.

As far as stand-alone shelves go - they are great and I've found that you typically can't build your own for less money.  What you pay for lumber and hardware/fasteners might be a little less expensive, but not by much.  You'll end up with a nicer custom product in the end, but not cheaper.