You do not draw a replacement card after drawing an Epidemic card.
You may discover a cure at any research station; the color of the city with the station does not need to match the disease you are curing.
On your turn, you can take a card from another player for an action, if you are both in the city that matches the card you are taking.
Researcher only: on your turn, you can take any City card from the Researcher (only), if you are both in the same city.
Your hand limit applies at all times: immediately after getting a card from another player, on drawing cards, etc.