Then determine how much you can afford to pay. If you haven't, check with a mortgage broker. Not just to see what you can qualify for, but how much you can comfortably afford.
Then you have the third number: $250,000, the list price.
The lowest of those three numbers--the comps, what you can comfortably afford, and the list price--is your maximum price. Don't pay a penny above that.
Then we get into the questions of seller motivation. (How long has the house been on the market? Does the owner have to sell?) And we get into your motivation as a buyer. How much do you really want the house? The higher the seller motivation and the lower your motivation, the lower your offer below that maximum price. On the other hand, the lower the seller motivation and the higher your motivation, the closer your offer might be to the maximum price.
Now, I don't know what's happening with real estate values where you are. Real estate conditions are local. However, you may find that, after the calculations I presented above, your maximum price might be, say, $225,000. That's a 5% drop from the asking price. But remember: In this hypothetical example, $225,000 is the most you can or will pay. Not a penny more. So your offer must be no more than $225,000, and it probably should be a bit less. How much less? That's up to you. Ask your real estate agent for some guidance.
Again, work through the calculations. Determine the maximum to pay. Then come in at or under that figure.
Hope that helps.
It does not matter if you pay via cash or lender, the seller stills nets the same, however with cash you can close earlier.
The county's appraisal is just a tax assessment. The comps from your agent are of market value. I would go depending on the rest of the situation (days on market, and whatever else your agent could tell you about the situation) with an offer of 2 to 3 percent below the asking price.