Dear TurboTax community, would it be a problem if my partner and I used the same phone number on our separate applications?

Do you mean on your tax return?  No.  

Dear Tom, 

Thank you so much for your kind answer. It was really helpful. 

Then, is it something normal that my application is still in pending status and hasn't been sent to the IRS since last Sunday, when I completed it?

It is fine to do that.