While playing Forgotten souls with 2 heroes (Knight and Rune mage), I had the Trash Heap room (where you have to search for the key) with one shadow dragon. I successfully found the key by the second turn, but the shadow dragon was still full health due to lack of surges rolled by the knight.

As soon as the exploration card is discarded (since I succeeded in completing the searches), you draw the next room card. Does that room's timer start as soon as you draw it , or when the door to it is opened ?

I still had a full health dragon to kill and only 3 turns to complete the NEXT room! it took me 2 turns to kill the dragon and ended up failing the next room due to time.

Was this done correctly? Should I have waited to start the next room's timer until the door is opened?

I am just learning the co-op version of Descent and played FS for the first time this last weekend and had plenty of questions...so take my guidance with a heavy dose of salt. But, I think you are a bit mistaken in your play. When you finish an exploration card, you do not draw another until you open the door. Instead, you have to advance fate and complete a peril card each turn that there is no active exploration card. In other words, as the requirements for the current exploration cards are winding down, start preparing to bug out for your next card. Tic toc.

Also, I need to check that specific card, but didn't that shadow dragon disappear when the search tokens were taken?

When you discard an Exploration Card, the next Exploration card doesn't activate until after you choose an action to open the door. When there are no Exploration Cards active you Advance Fate by 1 and Draw a Peril Card and resolve it's effects. Then you activate any monsters per normal.

Thanks for the clarification!