1 week is really short for a European city tour. You'd be spending much of your time traveling. I'd say MAYBE 2 cities you could do, but London and Paris both deserve at least a week of exploring, each. And as Metric said, it will still be pretty cold that time of year. But also cheaper.
Where are you flying from? I always take the length of the flight into consideration when choosing a destination for a short trip. If you're on the East Coast I would suggest the Virgin Islands, or Martinique if you want more of a cultural experience. There's also the Azores (Portuguese islands), which have gotten great reviews. If you can go on a longer trip (~11 - 14 days) I would suggest Egypt.
Let me know if you have questions about Europe, Egypt, or Asia - I've done quite a bit of traveling in those areas. I'm also planning a trip to Africa right now, so could probably point you in the right direction for West Africa destinations.
I'd do Brazil or Thailand. LMK if you want Thailand info since I spent quite a few summers there growing up. I might be a nutjob, but if you do end up going to Thailand, I'd recommend taking Thai Airways direct flight to Bangkok. My dad takes it regularly and he swears that it's so much better than having to deal with connecting flights in either Seoul or Tokyo.
The weather will be great in both places, especially compared to February/March in the northeast.