I am currently testing FormAssembly from Wufoo that we currently use but which does not allow forms to be as detailed.
Here is the link for the form test: http://www.tfaforms.com/223744
1 - Is it possible to build a rule on the Send button with different options of Drop-down lists (Department, quarter, year), to send to the correct department under these conditions?
2 - Is it possible that a form opens another form with the Send button, to force the first to be completed before another?
Using the Professional plan, both of those are possible.
1 - See: http://www3.formassembly.com/forum/discussion/469/send-email-depending-on-users-form-choice/ for a simple example. Your example seems more complex and we'd need a step by step description of how you're wanting that to work to say for sure, but at first glance I don't see why it wouldn't be possible with a little work on your part.
2 - Yes, I think this is covered under our 'Workflow' system: http://www3.formassembly.com/blog/support-documentation/managing-your-forms/workflows/ though there are a few other possibilities as well, depending on your detailed use case.
Again, both of those requirements would most likely need the Professional plan.