Sure. However, make sure to speak with your CPA before doing so to avoid any tax traps or unintentional gifts. You also would want to have a qualified appraisal of the property performed to set the transaction price.
Understand the transfer taxes, or stamps, that states/counties charge on sales, as you will be paying both sides.
I don't know if that's possible. That would be a refinance not a purchase. If own the LLC then you already own the property. I would ask the lender who approved you for the conventional financing that question. I think they would see it as a refinance too.