Unfortunately at this time TeamPulse does not have the ability to send emails to team members when a task or story is assigned. This is a feature that we have thought about however and it is in our backlog. The more users like yourself ask for things they would like to see in the app the higher the possibility that it be included in a future release, so thank you for the feedback.
As for your second question, your best solution for now while staying within TeamPulse may be to create a story and add tasks to the story that are the bugs that the QA team needs to look at. We hope to have a solution to problem/bug management in the near future.
Hopefully that helps a little. Another thing of note is that we do have our data service that you can tie into. You can find more information on that here
the Telerik team