I visited in mid September. I picked September because it's the most likely time to get clear weather to view Mt. Blanc. I had some nice weather but the first day was snow and it closed all the higher trails. I had to substitute lower trails for hiking.
I wouldn't plan hiking until July (you might get lucky in mid June) unless it's been a particularly light snow year.
The Aiguille du Midi should be open but enjoying it depends on the daily weather. It's pretty pricey. I'm not sure when the gondola to Helbronner opens back up? I saw it was closed but haven't seen anything since.
Burgundy is beautiful and not too far from Lyon. You could easily spend a few days in that area. I was there in early May and wished I had waited another month before visiting. The good part was there were very light crowds other than in Beune and Dijon. The bad part was it was still cold, the grape vines were barely growing and many restaurant choices in smaller towns hadn't opened yet.