When is it necessary to paint a wooden fascia (barge) board

Well I can tell you term for the 1X on the angled portion of the roof, it is rake mold. Most rake molds have a bit more elaborate profile, but there is no rule saying it can't be a simple piece of 1X. The term for the same piece running horizontally might be eave mold, but I have never heard or used it in all my years. This may have been a call made by the installing carpenter, for the idea of keeping things consistent, what is on the rakes, is also at the drip edge.

The blocks are there to keep the gutter from hanging at an angle under the roof edge so the rain does not over shoot the gutter, since it is fastened to the projecting 1X. If it was lowered, to eliminate the blocks, the rain when heavy enough, would shoot over the gutter, only light rains would be caught. If you are using aluminum gutters, you will need fasteners only 24-30 inches apart, unless you have heavy snow loads sliding off the roof, then you may need the fasteners every 16 inches. These blocks would only be needed at each fastener location. The composition roof may not let the snow slide, but the roof has a pretty high pitch.

You definitely do not want water running down the fascia board. The drip edge is suppose to run the water away from the fascia - wind can kick some water back onto the fascia but most water should be diverted away with the drip edge.

If the drip edge is installed correctly (over edge by about 0.5", under felt on the eaves and over felt on the rakes), you could try to get a thin piece of material between the drip edge and fascia to provide the spacing needed (you want about a fingers width). If you do that, be careful that the material you use does not wick water up behind the fascia (so make sure that the spacer isn't too close to the bottom of the drip edge).