Jason, parameters can only have one value at a time, which is why you had to setup two of them in your viz. Is there a reason you didn't just set up a Date quick filter? If you can then you'll have lots of options on how the filter will work.
On your second question, I'm not sure the YEAR(Date) is doing what you think it is. The map on the left is summing up 2013 Q2-Q4. The map on the right is summing 2014 Q1-Q2. If that's what you want then, yes you will want to leave the year in there. If on the other hand you want the totals for all the quarters in your date range, then you should take year off and get a single map of with all the totals.
Hope this helps,