The numbers you are quoting are all over the board and I am somewhat confused by that. Here are my quick thoughts to some of your questions.
The bank will likely will not get an appraisal, because they will hire an agent that is experienced. That agent will be able to complete a market analysis to price the home accurately.
The county website, if you are refering to assessed value, means nothing in the market place.
As far as what to offer is totally up to you and how you intend to use the property (rental or personal home). First, I would check with a local real estate agent to find out what the "sold" comps are in the area. Current lisings don't help much, you need to find out what homes are actually selling for and what condition they are in when they are selling. That will show you the "Market Value". There are plenty of posts here to explain what to offer for an investment property. You could use the same thoughts for a personal purchase too, but you are likely setting yourself up for disappointment if you don't get it. That is the risk you take when you take the "Low-Ball" route.