As long as there's any Contingencies still unsatisfied, yep, they can not clear it (aka, note as not accepted) and walk

We also budgeted an additional $3K for contingencies. 

A best course of action may be structuring the deal with the contingencies of a new survey and some kind of settlement with the neighbor. 

Either make closing contingent upon on the property being vacant.  

They say the "ultimate decision lies with HUD".While I have signed on their paper they have not formally ratified the contract and want to know if I am comfortable proceeding knowing this contingency.

You need to have working mechanical otherwise they will just do a larger contingency Exterior work can be done yourself unless they are not in compliance with HUD You can always go with a very low grade material for floor covering just to get through inspection.Lastly, some food for thought, unless you do not have the capacity in your budget because you are restricted by the loan or your DTI/Income, money is cheap and if you can have someone else do it now and you can place your time in your core job to earn more to be able to buy more, why not just let them do it.

Be prepared for unexpected changes with a contingency line item in your budget.We made a good profit on the flip, but I firmly believe that we did that when we bought the property.  

Yes, your quote should have been contingent upon building it a certain way, and upon receiving permits as such.

Lots of great connections all over but we also have a very strong and active contingent here in MA on here so make sure to find them (Set up some keyword alerts to find out about local topics) and get connected.Besides doing that on here make sure to get out to some of the many local investor events we have.  

Although she is family I would still have an inspection contingency on the property to ensure no major issues have been overlooked.