Part of the issue in terms of the high pricing is that you are making a lot of stops. UA goes to all the places that you wish to go on its own metal; ordinarily a multisector flight on one company's metal is cheaper than mixing carriers, but you're saying that is not the case. Or is it that the ticket is expensive because most of your flights are on CO (in view of the IAH/EWR sector) and the HKG/SIN is a CO codeshare actually operated by UA? Codeshare situations are still mixing metal.

First things first. Are SIN, HKG, and NRT each places where you wish to actually go and visit, or are any of these actually transits (ie just to connect a flight)? This might change what we might suggest that you do. If even one of these goes away as a legitimate stop, an all UA itinerary might just get a bunch cheaper.

One approach to this is to split your itinerary. You can check the prices of a paid ticket to go from IAH and some Asian point (eg NRT, HKG, TPE, BKK etc) and buy the ticket to the cheapest location. Then you can use an award ticket to fly within Asia. If we're talking about UA MP using Tokyo as your destination, it is possible to fly say NRT-HKG on say NH, then HKG/SIN on UA, then SIN/NRT on NH or SQ. Presumably you can do the same on CO OnePlus. Alternatively if your paid ticket goes to HKG, you can buy a cheap ticket on one of the LCC's such as Jetstar or I think Tiger to get to Singapore (this is what dd1612 is suggesting). Note that flying on an Asian LCC is similar to flying Southwest, you won't be able to interline anything (research this in the Budget Travel forum), plus they're much less generous with the weight allowance. Unfortunately AFAIK none of the SE Asian LCCs flies to Japan although I think that AirAsia is talking about doing this late this year.

To answer the first two of your questions (just because you asked; pogonation is giving you a lot of good advice):

1) Answered above. Note that at least recently, UA has been flying full loads SIN/NRT and SIN/HKG, plus end December is the end of the Singapore school vacation period - prime time, thus high prices.
2) For transiting through China, US citizens don't need a visa (however you need proof of onward transit, ie a boarding pass or an e-ticket receipt with the pricing information on it, not just a printed itinerary). There's a desk where China Immigration checks your passport and puts a stamp in it. There's also a provision where US citizens can transit in China for I think 48 hours but I've never tried it myself. In any case there's a big thread in the China forum on this.

Good luck!