The third week of escrow is indeed a critical time for any California escrow. The California purchase contract currently stipulates 17 days for the inspection contingency and 21 days for the loan contingency. So, both these deadlines come due on week 3.
These contingencies must be removed by these dates in order to avoid default. The buyer or their agent should send you a contingency removal form for each of these items.
Why is this important? For many reasons, the most important of which is that only after contingencies are removed do you have an iron clad contract. If, after contingencies are removed, the buyer fails to close escrow, the seller is entitled to a security deposit and can further sue for damages. Until day 17, the buyer can cancel the transaction for any reason and you are not entitled to a nickel.
One of the biggest mistakes that I see fsbo home sellers make is to NOT enforce deadlines, leaving the seller vulnerable from a legal and practical perspective. The is a California Realtor form called "Notice to Perform." If you have a broker, attorney or transaction assistant helping you with paperwork, this is the form to deliver to the buyer if the contingency removals don't come in. After you have filed this notice, you then have the legal right to cancel the escrow after a certain period.
Hopefully though, you will get contingencies removed, and everyone can move to the last stage of escrow. This final week or two is when the home seller should be handling things like termite repair and fumigation, as well as any repairs that were agreeed to. The buyer at this stage will be working with the lender to get the file closed.