Indeet, currently it is not possible to run any code from within IDEA. I'm not entirely sure whether you mean "running code like in Workbench", where you write a package and then you can evaluate the package inside Mathematica by pressing the run-button. Or, whether you mean evaluating expressions like in Mathematica, cell by cell.
The latter will never be possible, because IDEA is not meant to be a front end.
The feature of starting and debugging written code from IDEA is definitely planned. Unfortunately, I have no time at the moment to work on the plugin and this situation is very likely not going to change until the end of summer. Even then, I'm not sure how far I can take it alone, because currently, I'm doing most of the work by myself and I have only a handful of people who take time (from their tightly packed schedule) to help and to test the features thouroughly.
Initially, I had thought that there are more people who prefer to write code in an IDE when it comes to package development. On the other hand, I never had illusions about how many people are using Workbench for package development, but it seems that even my low estimates were too high. On Stack Exchange our user ID count is 13145 at this moment. If I should make a guess, then I would say from these people only 50-100 are using an IDE or an editor for Mathematica package development. That's not even close to 1%.
You may ask what all this has to do with your initial question. It's this: When I started hacking up the plugin, my idea was to take care of the core functionality and things I'm interested it (autocompletion and code prediction is one of my favorites). I wanted to provide the code freely so that all other people who (1) cannot pay for Workbench, (2) like to tweak things on their own without waiting for an official bugfix and (3) have a bit Java experience (I don't have much myself!) can take the plugin and implement all the fun stuff they like.
It seems my plan didn't work out and for me alone it is very much work, especially with all the other fun stuff I do in my free-time.
So to give you a final concluding answer: Yes, it is possible to run/debug a package. You just need to implement it :-)