Are bidders allowed to inspect the property prior to or after placing bids/having bids accepted? 

As of right now, the seller has accepted my offer and my realtor is drawing up the contract.

This is a side topic but I'm a bit uncomfortable with the thought of accepting them so I'm curious if you all would.

YOU NEVER KNOW WHAT POSITION THE PERSON IS IN AND WHAT THEY MAY ACCEPT.

Being relatively new to this, though, I don't know whether it would actually be a good deal for me (the current owner likely wouldn't accept it either, but I'd potentially be willing to try making an offer if my idea sounds reasonable).

Once appointment is landed, I can take a cash offer from my investor - if they can't accept for one reason or another, then I'd offer to list their property retail (MLS.)

On a serious note the pet's that you have on the premises as the owner doesn't obligate you to accept pets in your property, (though you may want to pet friendly rentals are always in demand) just know that having the pets on the property might make some renters think twice about living there if they don't like dogs, have allergies etc 

This makes it easy to accept or reject their application.

We have not closed on the original house yet we just accepted an offer today and closing August 10th so we are scrambling a bit.

Before you make the offer, ask the seller if he would be willing to accept a little less up front and monthly payments for the next... 3,5,10 or so years.