Should I ask about the lease or just wait?

We applied for an apartment a little over a week ago. Last Wednesday we got an email and a phone call stating that we got the apartment. I talked to the landlord on the phone and told him we’re still interested and he said he would send over the lease. Well I still haven’t received it by email or anything. I was wondering if it’d be okay to message him and ask him when we should expect it? I know the place won’t be move in ready until July 1st but I’d like to get the lease signed and things tied up. Would I be too pushy asking?

Also like I said the apartment won’t be ready until July 1st but I want to ask if we could move in some boxes and perhaps furniture the weekend before. Just because I’m pregnant and my partner works all week, so I won’t be doing any of the real moving. The weekend after July 1st is holiday (and my birthday lol) and it’d be nice to move some stuff in early. Of course we’d be willing to pay early rent for that week. Would it be okay to ask?