What does “Every Jewish mother will die, a fact many are not exactly famous for playing down” mean?

a fact many are not exactly famous for playing down

Sarcastic understatement. The implication is that Jewish mothers put guilt trips on their kids with statements like "One day I'll be dead, then you'll wish you could pay me a visit!"

The "would it kill you..." is over the top. It is for someone who has not done what is expected of them. Wife to husband: "Would it kill you to take out the trash?!"

I'll take a stab at Part 2 of your question.

As you might have guessed, Would it kill you to do X? is a bit stronger than asking Why don't you do X?

The implication is that a person is reluctant to do something for some reason – perhaps they are too busy, or they have some other reason that they've avoided doing something that really shouldn't be that big of a deal. So, the only reason you'd say, "Would it kill you to have a drink?" would be if such an outing was long overdue. If you and Bob had mentioned going out after work sometime, and several months passed, and it just never worked out, then you might say, "Gee, Bob, would it kill you to go out for a drink?"

It's often used mother-to-son to mean, "It's been so long since you visited me – do you think it's too much to ask for you to stop by and see me every once in awhile?" It implies the mother isn't being visited as much as she would like, and perhaps the son is acting ungrateful for all the sacrifices the mother endured while she was raising her son.

This article might be worth reading as well: Wikipedia entry on the Jewish mother stereotype.

The phrase

Every Jewish mother will die, a fact many are not exactly famous for playing down

may be doubly confusing if you're not familiar with Jewish mothers and English is not your first language. There are two tricks going on here:

• There is a figure of speech here known as litotes. By denying that they play it down, the author is affirming that they play it up.
• The Jewish Mother stereotype has these women guilt-tripping, in this case by playing on their age and infirmity to get others (e.g., their sons) to pay attention to them (say, to visit).