Wondering what sort of conditions to expect, whether there's snow, etc?

Yes, there is snow! I didn't have to cross much snow between 10,000-11,000 feet last weekend on the ridge from Johnson and Rafferty toward Merced Lake and trails were snow free below, but there is substantial snow still higher than that and on shaded/north facing aspects. Little snow fell early in the season, a decent amount of snow fell late, and late spring has been cool; thus snow is not very consolidated. If you climb snow slopes very early in the morning with crampons you may enjoy yourself; as the day progresses the snow gets really mushy with substantial postholing likely. I saw a Lyell report from a few weeks ago indicating the same situation and experienced similar conditions back then around Tyndall.

Here's a shot of the highest part of the Cathedral Range, including Lyell--

https://photos.app.goo.gl/27t95gjquwLT9yWx6