As long as you can legally enter the country they are not going to care that you have to enter Singapore for a few hours to claim and re-check your bags. You said you plan on spending time in Sydney so you are going to have to claim your bags anyways.

Looking at random dates in March it seems IAH-LAX-SIN ($1400) and SYD-IAH ($975) in Y would book as separate one-way tickets.. You could book on the same ticket for $4000 in Y, so don't do that lol

If you want to keep it on *A the SQ SIN-DPS flights are at 8:20am, 9:20am, 4:05pm, 6:05pm which kind of sucks as the UA flights arrive at 6:50 and 8:00am. Scoot has a 10:30am flight ($75) and Air Asia has 10:35am and 12:15pm. You won't get an *A flight from DPS-SIN. It seems that the JetStar, Virgin or Qantas flights are about $200 on that leg.