A hard money loan is probably your best bet. Quick execution, fewer hoops to jump through compared to banks. Traditional lenders typically won't finance your rehab costs, so depending on how much equity you are willing to put in, HMLs are the way to go. Bear in mind, these are usually short-term loans and you will need to sell or refinance with a traditional lender once the repairs are complete, depending on your strategy.
A fix and flip product would make the most since for sure, however the type of loan you should go for really depends on the scenario .
I'd reach out to a local mortgage broker to discuss your options. Someone in my market was able to get a 25% down construction loan from Wells Fargo through his mortgage broker. Unless you plan on living in the property, you'll have trouble finding a construction loan with less than 25% down.
As for the contingency budget, most investors have their own opinion based on their experience. I usually add an additional 15% to the total renovation costs as contingency, but that is for value-add deals. If you are looking to do a full rehab, I'd assume higher than 15%