But when I make the offer, and when/if it’s accepted, do we go straight into taking about the contract?

@Jason Merchey In my experience, we've always accepted a signed lease regardless of age (+ proof of deposit if its brand new). 

Just about five years ago I was finishing up graduate school at Texas A&M and had just accepted a job to teach high school and wrestling in San Antonio.

Also, if it is not permitted, get it in writing from the agent that the company will accept it.  

If they meet all that and the references check out, it’s an automatic application acceptance and I present them with a lease.

The paperwork also says that consent cannot be unreasonably withheld but I'm not sure the reason I have for wanting to move is an acceptable reason.

Also, a great next step if/when your offer gets accepted is to open an LLC if you have not done so already.

If the seller accepts and I buy it for $50,000 and sell it for $100,000 it’s a free market transaction.  

You should be able to get the tenant to sign an agreement that in exchange for the two extra weeks they will leave by end of that final day.Waiting to see what a tenant is going to do can be really aggrevating, but the eviction process should be a last resort, and shouldn't be started if you've accepted rent. 

@Daniel Pierson not local but PenFed will accept heloc's on investment properties