My ex and I had real estate licenses, in California, years ago. I also researched real estate investing opportunities. The real estate laws vary some from state to state. Each lender also has their own requirements so it's hard to say exactly what's allowed in this case, especially since we don't know what state they're in.
Pre-approval is the wisest way to go. Then you know what you can and cannot afford and you're less likely to have expensive delays.
The deposit sounds reasonable.
Paying off their car loan reduces their monthly obligations. Lenders have formulas for what percentage of your monthly income you can afford to commit to payments on large purchases such as cars and houses. Paying off the car loan gives them more room for mortgage payments.
Having a required amount in the bank gives the the appearance of being able to make so many mortgage payments even if there's a loss of income.
The debate here seems to be about the extension fees. Your sister entered an escrow agreement before securing financing. The financing is taking extra time. Your sister now needs the seller to hold the house for her. The seller is wiling to do that, until the end of the month, then he's been advised to charge a daily fee if they need another two weeks. (...the owner changed their mind saying he will give her and my brother an extension until the end of this month but they have to pay an extra $100 per day after for two weeks) This is incentive to get their loan approved quicker. The seller could be renting the place out, but it's sitting there costing him money instead of making money. He may have other buyers or simply need to get on with his life. He may need the money for something, like a house he's buying, and the delays could be costing him money or worse yet causing him to lose out on an opportunity. We also do not know how long this has been going on so the seller may be tired of all the delays. He could have been ready to quit the deal because it's dragging out so much, but the addition of the fees gives him incentive to be patient a bit longer. It's a stressful process on both sides of the deal.