Buying with contingency of selling current residence

[Account Closed]

Realtors, bankers, and the BP community with experience with buying and selling primary residences simultaneously:

My parents are looking to downsize their current home. The county's assessment for the home is $430k and other similar properties recently sold for upper $400s and $500s in the neighborhood. They have roughly $80k left on the mortgage (<5 years left). The area they are looking to buy has properties in the $300s. 

My parents want to either put in a contract to buy with a contingency that they need to sell their current residence first; or, sell their current property and rent (storage and living) while looking for a new home. We haven't spoken to a Realtor yet and one of the many questions for the Realtor would be how a transaction like this would occur. 

I would like to ask for the community's input on the following:

1. If they sell first, they would pay off their mortgage and the remaining (post closing) amount is theirs to, for example, put in a cash offer on the new home. Where and how should they store this money prior to making the purchase? They would also like access to this fund to cover the rental (storage and living) costs until they can make a purchase.

2. What are the pitfalls that they should watch out for either buying then selling or vice versa? Mistakes Realtors, settlement company, sellers, etc. may make.

3. My parents and I could pull our funds together and pay off the mortgage. For simplicity, let's assume that whatever fund I contribute would be a gift (with consideration to the federal gift tax) and there would be no repayment. Would there be any benefit to paying off the current mortgage first?

Thank you.