Will this new Cold War spark another space race?

Military space is pretty likely to kick up IMO. Space contractors will point at this situation and say, "see, all the money toward simple space-leveraging systems is worth it because it keeps your boots off the ground".

I could see an energy race or big ramp-up, particularly around demand for reducing cost and vulnerability of energy systems.

It's already ongoing to some degree, but nations around the world can now see very clearly how their energy systems are of direct interest to their adversaries in conflicts. Adversaries looking for easy bargaining chips will attempt to capture or take down power systems, casually suppressing meaningful civilian activity while also thwarting a lot of defensive information insurgencies. (I'd be interested to see stats on the spread of viral content under various power/infrastructure circumstances, because a lot of this content is essentially in a race with other viral content to win info-battles)

This development could then kick crypto mining into very high gear.

It's interesting to me to see that we may be at a kind of threshold between the years when it was still possible to place your adversary back in the early 1900s technologically, and the years when such a thing will be very hard if not impossible, and will perhaps even become a war crime.

Edit: Regarding Mars, I'd say the situation more importantly/more likely could set us up for a long-term sustainable "relationship" with Mars but also the entire solar system, particularly if we can start to demonstrate and make use of meaningful presence on the Moon. It starts with efforts like Artemis / Gateway.

https://www.nasa.gov/topics/moon-to-mars/

The space race was a friendly way to tell "I have better ICBM than you.". Now everyone has ICBM, so it's not important to make a show about it.

I don't think so. None of the pieces are there, Russia has no money and it's brainpower is leaving.

The rest of the world is doing just fine and in a space race themselves.

I'm not an expert, but given the fact that Russian space agency officials have threatened to bring the ISS crashing down to Earth to destroy it and any city in its way, and other countries are scrambling to figure out how to continue without Russian equipment in as quick a turnover as possible... I doubt it. I think if anything, Putin's latest tantrum is just going to slow down, or even temporarily halt, space exploration.