I am headed to the Dolomites around April 7. I have heard many places are closed. Is that true, and if not, are most open and is it ski or hiking season?
Here you will find some information about skiing - some lifts are open, some are not - but either way around that time the snow is likely to get slushy mid-day. Not impossible to ski on, but something to consider.
When you say "many places are closed" are you referring to recreation, lodging, restaurants? I haven't been at that time so I can't comment on it but clarification may help others better help you. The town Bolzano seems to be pretty year-round, you could always stay in a place like that and then explore to what is open at that time.
What's closed? It can snow in April and many ski resorts are still open.
In Ricks Italy book he writes,
"Spring is usually dead, with lifts shut down, huts closed, and the most exciting trails still under snow. Many hotels and restaurants close in April and November."
Were you hoping for some fun indoor activities or outdoor activities in the area?