Welcome to Qlik :-)
Answers in order of your question:
1. Add a bar chart and then add your Dimension ("Case Category" maybe?). For measure, qlik measure and choose Count of "Case Category". Here is a quick 3 minute video on Intro to Qlik Sense that includes how to add a Dimension and Measure.
2. There is a pivot table Visualization. You can Add Case as a dimension. Your average and number of cases can be added as measures.
3. You can upload Qlik Sense applications to Qlik Cloud and invite up to five others to view your applications. The people you share it will be able to interact with the charts just like you would on desktop.
Qlik has a very powerful Calculation engine and all that you are asking for is totally possible as Asim already mentioned.
You have full command over the data and its calculations.
Apart from that, Qlik even allows you to visualise the data the way you want.
Just have a look at the kind of visualisation extensions which have been built in QlikSense here: http://branch.qlik.com/