Well, you have a couple of issues first of all. If you want to refinance at the value of the home, you need to wait 1 year. Fannie Mae, Freddie Mac and FHA and VA will use the lesser value of the sales price or the appraised value. So, in your case even if an appraiser told you the house was worth $400,000, the lender will give you a value of what the sales price was. You need to wait 1 year.
If you want to refinance to lower your rate only, you need to wait at least 6 months to go FHA. Why FHA? Because if you wait a little while longer, you can do a 203K loan through FHA. This will allow you to refinance the home and add up to 35K or 110% of the after repair value, whichever is less. It's almost always the 110% numbers used. So, I would recommend finding out how much it will cost to repair or upgrade the house. The appraiser will do 2 appraisals and you will have to pay for both. The first will be for the value of the house (65K). The second for the value after repairs $75000 for example. So, if the house is worth 75K after you repair it, then you can borrow up to 82500 which equals 17500 in repairs. You won't get all that money as the lender will charge fees for the loan, escrow and 203K fees. You may have about 12000 left depending on the time of the year you close due to escrow reserves needed at closing. The contractor needs to be licensed, insured, have references, and a resume. He/She will be paid half at closing, in most cases, and the other half after repairs. The 203K loan gets complicated when you borrow over 16000 as it requires HUD inspections when you go over that amount. The loan process can take up to 60 days depending on your contractor's ability to get things to me as quickly as possible. One loan I'm working on right now, the contractor has delayed by 1 week because he was too busy to get me the information I needed to complete the invoice requirements by the bank.
With regard to your debts, you should never use your home equity to pay debts off. There are better ways to do that and I can help you with that at no cost. No, I can't pay them off for you, that takes all the fun out of it.