I thought the conventional wisdom on the board was NOT to put the capital on the precursor tile, since it's the %age of what it was already being produced on that tile, and the capital produces no tp on its own.  I think I remember hearing that putting a capital on a precuror tile gives no bonus at all. 

I have no idea why the summary show 738tp.

The good numbers folks around here will want to know more about both the colony in question (Colony window screen caps seem to help folks get useful answers) and your general econ context.

I try to read and learn from the smart numbers people, but I'm not one of them so I can't give you a useful answer. But, I can imagine everything from you having found out that the Tech Capitol actually produces research directly when most of us thought it didn't to you've just noticed what an insane basic research level you have empire-wide (big stack of permanent research ability bonuses plus a huge collection of research mines?). Or it could be a bug that's very hard to reproduce.

Here's a cap of the full planet after the tech capital was added:

I think the problem is that the improvement summary screen is incorrectly reading the +100% research on the tech capitol as +100 research points; that is why you are getting an extra 700 points showing up there that aren't showing up anywhere else. Compare the tech capitol to the lab below it - the lab produces 12 tp and gets a 12 tp bonus (correct) while the TC makes 100% bonus but no TP production (correct) and is getting 700 TP bonus production (wrong). The main planet screen is calculating correctly.

This has been tested out fairly well, only the political capitol gets any bonus from a bonus tile.

The political capitol is an anomaly because what it boosts is handled differently from the others. Labs, factories, and farms (included because there are % buildings, even if there isn't a capitol) all produce units of manufacturing, research, or food. The bonus tiles for these commodities boost structures that produce points, but not those that themselves produce % bonus.

No influence, morale, or economy structure produces points, they ALL produce % bonuses. So limiting those bonus tiles to boosting "point" production for commodities that don't get produced in "point" form would be silly (econ bonus tiles don't exist, but since they would theoretically fall in this category I included them). That is why the political capitol is an oddity, only it boosts things that can be boosted by bonus tiles - and it can benefit from either type of bonus. If econ bonus tiles existed, you could put either the econ or manufacturing capitol on one to get the bonus.