Planning a trip there - how many nights do you suggest?

Summer or winter?

1 night minimum but you could easily spend a week or more. Lots to see if you do some hiking. Mineral King is pretty cool and it was almost turned into a ski resort by Disney!

You need 1-2 days to do the big trees, then lots and lots of things to do in the backcountry. I haven''t been there in years but it''s pretty much classic Sierra Nevada scenery. I do not think its a good winter place, it lacks the year-round infrastructure Of a place like Yosemite.

I''d shoot in Sequoia twice, before I''d shoot in Yosemite once.....assuming your planning on doing model shoots there.
Pure vacation is another story, but similar in certain ways.

i'd spend weeks in yosemite, if i really wanted to see it all. but that would be backpacking trips. if all you want to do is see the major sights in both parks, you could do that in four or five days. and go in either the fall or winter. the summers are just too crowded. the best times i've had in yosemite were between columbus day, and memorial day.

We stayed there a couple of nights at the Wuksachi Village in the late spring.   Dining room was pretty good too.  Both are expensive!

Probably only the main park road will be open and plowed, The Generals Highway.  The road to Kings Canyon likely to be closed for winter and probably a lot of snow gates will be locked to places like the meadows or Hume Lake, Morro Rock, and the two caves too (One in Sequoia and the other in Kings Canyon.).  I would bet on a lot of limited access in winter and parking along the main road will likely be non-existent if plowed.  It's like driving in a tunnel with no exit or pull off anywhere if it is packed into an 8 foot wall along the roadway - you just keep driving until you get to a major business point in the park.  There were times in the winter where the fog and clouds and snow on the trees that I wanted to stop and photograph, but you couldn't stop anywhere due to the long single-car caravan.

You probably could get by on one night for winter if you are looking for dusk or dawn photos due to the limited access.  Summer offers much more access and sights, imho.

Fwiw, Death Valley N.P. seems to offer more opportunities in winter and weather is nicer.