How likely is it that Discovery went to a mutable future, just one of many, especially with the Temporal Cold War, Carl, Q, Trelane, Janeway, the HMS Bounty, and any number of other temporally active agents out there in time? How locked in is the 32nd Century?

About as locked in as any of the Time Travel in the 23rd and 24th centuries.

Star Trek time travel can be inconsistent, but usually, it tends to stick with there only being one timeline that alterations shift back and forward, something that isn’t really helped by the Time War.

The only time that we’ve seen anything approaching an alternate timeline like that is with the creation of the Kelvin timeline from the Narada incursion, which resulted in bidirectional effects that separated it into a new, independent timeline, but events like that are more the exception than the rule.

Though, normally, Trek time travel rules would suggest that anything lasting longer than a season (or into the next episode) is usually here to stay, if it’s not reverted at the end of a multi-parter. Data’s head remained centuries older than his body, for example, and the crew of the Bozeman are still rattling about the 24th century, having jumped 70 years into the future.