I would expect that a significant contributor to the surge prices is from HVAC units and similar needing to work harder/etc.

That's one end of the equation. But the other end is in how we're replacing coal plants with natural gas plants.

Coal plants are significantly slower to respond to market demand (on the scale of hours to increase/decrease supply), so they need to be run at a higher output on a longer time frame as electricity demands rise. Because ERCOT auctions electricity demand in 15 minute intervals, coal plants can't meet a short spike in demand before its come and gone. Natural gas plants don't have this problem. They can sit on their reserve fuel until the prices peak and then flood the grid with electricity on short notice.

As coal plant profitability sinks relative to gas plant cartels, the volume of electricity we produce becomes more and more easy to rig within the ERCOT auction markets. HVACs going into overdrive in the evening (typically between 3-7pm) signal a potential spike in demand. But gas plant operators get to wait until the electricity auction realizes those high prices, rather than producing electricity in advance and hoping you get to ride a wave through sunset.

do solar panels work better in warmer temperatures and does output of solar panels scale anywhere close to comparatively with ambient temperature and/or need for HVAC and similar systems?

A lot of the heat in cities like Houston comes from the humidity combined with the sun, so a bit of breeze can drastically impact the gross demand for electricity. Meanwhile, electric components of all sorts (photovoltaics included) perform worse in the heat. Breeze can also impact electricity available from wind turbines, which further shift prices.

Batteries can help renewable energy companies hedge against peak production relative to peak consumption. But, again, a private market maker still wants to chase the highest returns. So putting a bunch of quick-to-discharge batteries on a grid alongside quick-to-ramp-up natural gas turbines means... more cartel price fixing.