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The problem with saying "don't take x route because of weather" is that there is no way to get across the US by car without the risk of winter weather. Even going down to I-10, seeing snow or ice is a very real possibility. I-40 does stay at a higher elevation, especially out west, so the chance of seeing winter conditions are a little higher along I-40, but the only way you'll know which is better during your travel times, is to look at weather forecast just before you leave.

In this case, the difference between taking I-40 and I-10 isn't very much either way, so make plans for either route based on the actual conditions.

500-600 miles is pretty much a full day on the road, and that's about as far as you should plan to get. If weather stays in your favor, that means you'd be looking at 4 long days, but you should plan at least having a 5th day available in case you have to slow down or stop due to a storm.

You really don't need to plan where to stop for gas, as it is rare that you'll have more than say 50 miles between gas station. As long as you are looking to fill up by the time your car gets down to the 1/3-1/4 mark, you should be fine. Same with cell phone, there is no way to say where you'll have gaps in service, as it depends upon your carrier/phone. Reception is usually pretty good along interstates, but it is not universal and you will have patches where it is not available. That said, people have been taking roadtrips for decades before the cell phone was invented and you'll never really be alone when you are on an interstate highway.