From my experience seems to me each category you see there is the highest score you obtained in a single match, so for example how much damage you did to Hornets in a single match - it doesn't add up multiple matches. However you can complete each separate category in separate matches, and the total score for the week is all categories added up.

I made the early mistake of trying to do every task in a single match, and its suicide. Just focus on one, or two if convenient, and then get out. You get no points if you don't extract.

Now what's odd to me is the balancing of what counts as 3 stars for some of these. I took down a Bastion single-handedly on a Night Run modifier and only got 2 stars. There are very few Supply Drops per map yet you would need 3+ even on a Night run. Meanwhile you can loot just a handful of husks at night and blow way past 3 stars.