We have come to think of the inspection period in the housebuying continuum as the second negotiation. If professionals do thorough inspections of homes, they can almost always find things that are wrong, or that could go wrong in the future. In fact, when we bought our home 25 years ago, so many things were given a "D" condition rating that it's a wonder we could live there--and some of them have never needed further attention. On the other hand, many things needed fixing, some right away, that weren't found in the inspection.
So, let's say that you are the buyer, and you have ten days to get all the tests you want. You should get a non-relative professional to check the property, and you will then receive a written report. Keep careful track of the dates, because your right to withdraw without penalty expires at a certain time, if you do not give written notice. Obtain estimates for fixing the more serious items mentioned, and think about what you really want to ask for from the seller. In today's tight market, there could very well be a backup buyer, and you are going to lose the house, if you ask for too much. Remember that you are looking for defects, not aesthetic choices. The fact that you'd prefer different appliances does not make that an item for the seller to address.
Ok, now you have a list, with costs. Put the things you feel strongly about into a list, that you keep as short as possible, and send it to the seller's agent before the date of expiration. Write it so that you are excused from the contract unless agreement on these items is reached. In our company's contract, we put a limit on that time, and say that the entire contract is null and void, if agreement on the inspection items is not achieved with seven days of the end of the period. That limits the chances that negotiations will drag on, and that one party can feel that there is still a contract, while the other is ready to move on.
The sellers will often counter with a list of what they will do, and what they will not. At this point, you should consider the urgency of any repairs, the structural nature (or not) of those repairs for safety purposes, and any work you plan to do, which might make those fixes moot or cheaper. Consider other ways to resolve the differences, from having the seller throw in furniture or equipment instead, or giving a lump sum of cash, with you assuming the repairs yourself (which will guarantee that they are done in the way that you prefer). At the end of the day, it's a gut call about what you can live with, and how much you want the house. Again, pay close attention to the time remaining to decide. You may have time for another round of negotiations, or you may not. Don't insist on things that you will later say you could have given up, unless you are sure that you are prepared to walk away, if you get a negative response from the sellers.
The same is true of the sellers, with the additional caveat that you should agree to any repairs that you will have to make in order to sell the house to any buyer, meaning that you will have to do it for the next person, if you don't do it now. Remember that you will have to disclose any flaws to other buyers, if they are indeed problems. Don't let a few repairs stop you from selling to a good buyer. Time is money, and you will have to start the whole process over. Don't get fixed on a number that you want to receive, because this is a business decision at this point.
To both parties: Keep your eye on the closing!