What’s the best time of year to throw big swim baits?

Anytime is a good time to throw, but my favorite is during the shad spawns. I always have several tied on.

Water temps below 70, cloudy, with a little wind

Yes. My best big swimbait bite starts early April and goes until mid August.
April and May are the best for shallow big swimbaits. Some days just after the LMB spawn they want them more than a senko.
*I consider the 10XD and the like as deep diving swimbaits—start throwing that again mid May

I don't always throw swimbaits but I major in them...and I have for more than 30 years. I can't tell you the "best" time of year because I don't fish your lakes but I can tell you I have caught DD bass on swimbaits in 10 of the 12 months of a year. There are no months that I don't have one tied on and I'm not counting paddletails.There is certainly times of the year on certain bodies of water that I'm focused on throwing DD crankbaits and Bama rigs but I'm always ready to exploit a swimbait bite that virtually always exists.

After the spawn

I honestly didn't think there was a " bad" or wrong time to throw them. If your waters have shad, gills, etc then they will work year round.

What’s the coldest water?

45 degrees or so is about the coldest water I have thrown them in.

I like to throw them in May into early June.

March-May on a clear water big bass lake is ideal. They can work year round but that is when I’ve done the best with them.

Anytime if you're fishing for a big fish. Nov/Dec if you're fishing for a giant.