Let them get through several frosts before covering. You want the ground to be frozen solid before you cover them. You want to keep them frozen all winter, the covering of straw is to keep them cold. Otherwise, freeze-thaw cycles will hoist the shallow plants out of the ground. Uncover them in the spring a bit before the last frost, but keep the mulch handy in case the early flowers get caught in a late frost.

It's probably too late to thin them, that's best done in July or August. I think thinning might stress them too much before the winter, and there isn't a lot of time left to overcome stresses. Don't worry too much, they'll still produce fruit!

I wouldn't worry about mowing them, the frosts will kill the leaves eventually. I think the extra "fluff" from leaves actually protects the crown of the plant, a bit.

Thanks - should I cut off the runners that haven't rooted yet? Thin in the spring (and maybe transplant the excess/daughter plants)?

I don't have straw, just old weathered hay and burlap. I was thinking burlap might be best to put down first, so I don't have to pick hay out of the bed in the spring?

The hay will mat down over the winter, you'll want to keep a bit of mulch on it to keep the strawberries out of the dirt.
Probably thin next summer after the fruit is done. Otherwise you might spur the plants to grow, rather than fruiting. I imagine whatever runners haven't rooted will be killed by frost, so you can save yourself time and not worry about them.

The strawberries are mulched with bark mulch right now, I pulled it aside to root the runners, but a lot rooted themselves, I guess it decomposed over the summer. I was thinking burlap over the plants and hay over that, remove both early April and put down new mulch around the plants to keep berries out of the dirt. Though some spots are so thickly grown I can't even *see* any dirt!

Or do I need to get some kind of mulch down in between the plants in the next month or so?

No, your plan sounds fine.

Great, thanks!