I've got a Felt X-City with the Shimano Nexus 8-speed internal hub, so I need to adjust the chain tension through the bottom bracket. It seemed easy but I am having a problem, so maybe I'm missing a step. Here's what I'm doing:
1) loosen the 2 hold-down bolts at the front of the bracket (shown by the arrows in picture 1)
2) use a spanner to turn the bracket until chain tension is at desired level (using the 2 holes indicated in photo 2).

But the spanner tool, which fits great, won't budge. Should it be that hard? Or do I need to use 2 spanner tools at the same time, one on each side?

The LBS adjusted it before, so I know it can be done!

Might be an Issue of corrosion in the inside of the frame BB eccentric and BB frame surfaces.

, then you have to take it Out entirely and clean off the surfaces and apply grease

or grease based antisieze on those surfaces so It will work better for a while.

"until chain tension is at desired level" - which should be no tension. The chain should be able to be moved 1/2 to 3/4 inch when the chain wheel is in the position which results in the tightest chain. Any tighter is pointless and will result in more chain and gear wear.

Try putting a hex wrench or other sturdy rod into one of the holes and turn the crank until it contacts the rod, then use the crank to rotate the BB.

Either you have a corrosion problem from water entering the BB, or possibly when the set screws were tightened they raised burrs which are jamming the eccentric.

The tip-off as to which is whether you can move the eccentric slightly. If there's zero movement, it's likely corriosion, but if it budges slightly then likely a burr, and some extra force will do it.

I move things like this not by way of a pin spanner, but by holding the largest bar which fit's the pin holes into one, and levering with a small crow bar between the inserted rod and the spindle.

As pointed out above, the desired chain slack is such that there is still visible slack at the tightest crank position. If it's rigth you can still rotate the crank slightly to visibly transfer the slack from the bottom to the top loop and back. Since chainrings are rarely (if ever) perfectly round, make sure that the chain never gets tight as you pedal.

If it is corrosion, you might start by pouring some ammonia down the seat tube, and letting it soak for 24 hours. Once you have it moving, remove it entirely and coat it and the shell with grease before inserting.

Measured along the top of the chain or the bottom. Or does it matter?

It's the same. It's a question of total slack. The surest way to be sure is to hold the rear wheel and make sure you can transfer visible slack from one side to the other. I think the 1/2" figure is way too much, and the goal is minimum slack but enough that there's still visible slack at the tightest point in the crank's rotation.

It's def. seized. Here's my plan (in order):
1) ammonia down the seat tube (really?) ($2)
2) let LBS have it overnight while they soak it in PB Blaster. They are also getting in some tool that freezes seized metal parts to aid in separation. ($25)
3) force it out and replace ($105 + labor)

Are there any other problems caused my a seized EBB other than chain tension, because I'm not noticing anything?

No there's no consequence to ignoring a frozen cam in an eccentric BB if the chain slack (tension) is correct. In some cases it might get more frozen over time. The more serious issue is that your BB might get frozen into the eccentric, making maintenance or replacement difficult if not nearly impossible.

You can continue to ride it, or you can try to free it up now. The easiest way to free these if they're not too badly corroded is gently spread the shell a few millimeters by putting a place between the ears and threading the bolt from the wrong side so it presses the ears apart. (Edit, I just looked at the photo, and you're isn't a classic split shell with pinch bolts, so the preceding doesn't apply, I leave it there in case it helps someone else).

Then dolly the frame on a support that fits the frame's BB shell with clearance for the eccentric, and drive it straight across with a well placed hammer blow. Or, if you have a friend with an automotive shop, his bearing press is perfectly suited to this task, and will pop it free quickly and easily, probably with no damage, though you might have to buy a proper support ring.

Once the eccentric is out, clean it up with emery, and likewise the shell, then replace using grease or anti-sieze to prevent a repeater.