Yes, you have 7 days to ship an item after a claim has been made. So even if you only shipped on Mondays, even a claim made the prior Monday would still be within the 7 day time frame to ship.
If you mean non-flat packages then unfortunately no, photos of stamped packages are not acceptable as proof. According to the lettucecraft shipping rules packages are required to have tracking when shipping. The exception to this if for flat items that the post office does not allow tracking on. If your item is a flat paper item (example an ATC card mailed in a regular greeting card) and can be mailed as a 1st class letter, then yes you can send it as 1st class letter without tracking as long as you agree if your letter goes missing you will remake, resend or refund the points. All other non-flat packages need to have tracking.
Swap rules for lettuce craft are here:
The part I’m referring to are these quotes:
~ Mailing your swap items to your swap partner
For packages sent within the U.S., you must use package tracking when you mail your package.
Exception to the tracking rules: some shipping places (like the US Post Office) won’t do package tracking/delivery confirmation on packages that are too thin, such as normal envelopes. Because of this, papercraft swaps are exempt from the tracking requirements, BUT participants are required to resend if for any reason their package is not received.
You can buy postage labels with tracking online from your home from the USPS or other places like Pirateship, Endicia, stamps.com etc. for first class and priority mail packages and they will provide tracking numbers. You just buy and print out the postage and tape it to your package, and then drop it at any USPS blue mailbox. Some of those sites listed require a monthly member fee (endicia, stamps.com) but the USPS does not. And while I have not used pirateship yet, I don’t think they require a monthly service fee either.