What's a good offer? Easy. Forget listing price. Forget price reductions. Forget tax assessment. Forget taxes.
Two numbers: What is the house worth? What repairs does it need?
That's it. simple. Accurate. Easy.
Have a Realtor do a CMA on the property. That'll tell you how much the property is worth in reasonable condition (reflective of other properties that sold recently).
Then subtract from that number the needed repairs.
You'll be left with a number. Let's say it's $200,000. Fine. Simple. That's the most you pay for the home. You might offer less. Your agent can provide you some guidance on negotiating strategies.
You ask "Will the seller's lending bank take any reasonable offer?" Who knows? More to the point, who cares? You make an offer that works for you, using the formula above. If the bank approves the offer, that's good. If not, then either the bank will have to reduce the price again or else someone else will come along and overpay for the property.
You ask: "The value on the county books is what percent of the house's real value?" It depends. In Fairfax County, where I am, the county considers an assessment accurate if it's within about 10% of the real value of a property. So a property could be assessed at $200,000, and the county would consider the assessment accurate if the true value were between $180,000 and $220,000. Now, that's for an ACCURATE assessment. There are plenty of inaccurate ones, too.
I don't know if Jefferson County has the same policy. It's probably something similar. But: Who cares? So what if the property is assessed at $200,000. It might mean the property is worth between $180,000 and $220,000. Do you really feel like paying $220,000 for a home that's only worth $180,000? No, I didn't think so.
So: Find out what the comps are. Subtract repair costs. That's the most you should pay for the home. Perhaps offer less--ask your Realtor for advice on that.
And forget listing price. Forget price reductions. Forget tax assessment. OK?
Hope that helps.
What does your Realtor say?
You have some great advice here, what you really need is someone that has seen this property, also knows the local market values and trends. Are values going up or done? What have comparable homes sold for recently?
Your Realtor's job is to take the mystery and uncertainty out of purchasing a home. If you do not have one, then find a competent Realtor. They will save you time, money, and stress.
The other answers posted are dead on. Make sure you have a time that the deal must be accepted by the third party (lenders). You generally will get an okay deal in the end. You do not stand to save a lot of money buying the property. The seller is the one who benefits from the sale. They walk away from the promise they made to repay the bank. Rely on professionals you hire only. The REALTOR, the home inspector and the appraiser are on your side.