Occasionally, there are notices/e-mails sent out regarding a seriously ill person, requesting that people say Tehillim for them. If the ill person is a stranger and there is no way to find out whether he/she has recovered (or, chalila, has departed), how long should one keep on saying Tehillim for them? Would it be correct to set a time limit?

It's a good question and I myself encounter the same problem. I don’t think you can say there is a set time limit, it depends on the circumstances. Best will obviously be to try to find the information about the person you are praying for, if not you can try to asses the average time this problem would normally take, this can be sometimes over a year and sometimes only days.