Is there a metropolitan card that would cover all (or most of) London transport, underground and buses, for one week? Could such a card also include transfer from some airport (Stansted, Luton, Southend)?
You can get a 7-day travelcard which is valid for buses, the underground, DLR, and almost all National Rail services. If you want it to cover the entire Greater London area, it is £69.80 for a 7-day all-zones card.
It will be valid for underground and TfL Rail to Heathrow (but not on Heathrow Express except for transfers between terminals 2/3 and 5 which are free), and of course for London City Airport.
Travelcards covering more than one day are electronic tickets that must be loaded onto an Oyster card. There's a £5 deposit for an Oyster card, which you can get refunded at the end of your visit (which may however involve waiting in line).
There are cheaper travelcards that will only cover the central areas of London (starting at £35.10 for zones 1+2, which will reach all of the major tourist destinations). These will not in themselves be valid to/from the airport, but if you use the Oyster card to pay for the airport trip using "pay-as-you-go" money, the system will automatically select the right extension fare to/from the boundary of your travelcard zones.
However, for the vast majority of visitors, travelcards are not the most economical option. Instead get the Oyster card, load some pay-as-you-go money onto it, and rely on the automatic daily fare capping to keep your expenses lower than a travelcard would be.
Unless you're going to spend your entire week flitting around in public transportation (not a bad word about that -- it's what I do when I'm in London, but I'm a railfan; it's what I go for), PAYG fares with daily caps will almost surely be the best option for you.
You can use the Oyster to and from Gatwick in addition to Heathrow and LCY. (Also Luton later this year; see comments).
A final option, if you have a recent-ish international payment card that supports contactless transactions, is to use that to pay everywhere you go. Such cards are accepted out of the box by the Oyster readers in buses and at stations, and you pay the same fares -- and the same daily caps -- as for Oyster.
With contactless payment, your card will be charged once per day, so beware if your card issuer charges fees for each transaction. On the other hand you won't have to pay the £5 deposit for an Oyster card, and you won't have to worry about leaving balance on the Oyster you don't get to use.
Stansted, Southend, and (as of this writing) Luton are all too far outside London to have local transport with integrated ticketing. (You can buy train tickets from these airports that include a single-day travelcard for London, but then you'd still need to cover the rest of your stay).