where to stay in cortina and hiking suggestions
You might want to repost this under Destination Q&A, Italy to get some responses. You now have your question in a section where people give reviews of hotels.
I've never been to Cortina, but I assure you I had a unique experience in the Dolomites at this wonderful refuge:
https://www.matteoarghiro.it/rifugio-baita-belvedere-monte-rosa-valle-daosta/
We're on Monte Rosa in the Aosta Valley. Maybe you'll like it and find it useful.