If your going to use a herbiced I would spot spray only. (read the label carefully) Proper management practices that encourage a dense, thriving turf are the best method of weed control. Healthy turf shades the soil so sunlight can't reach weed seeds ready to germinate. A thick turf also minimizes the physical space available for weeds to become established. There are several management practices that will promote a healthy, dense grass. If turf is over-or under-watered, over-or under-fertilized, or mowed too low or too infrequently, the turf is weakened and cannot compete with weeds. Damaged areas resulting from using unsharpened mowers increase time needed for turf recovery, allowing for weed invasion. It is very important to understand that weeds don't create a void, they fill a void. Mulch Mow High 4+" with a sharp, balanced blade Water deeply and infrequently with 3/4-1" water. (no more than once a week in summer) What variety of grass do you have? When was the last time you fertilized?

I recommend getting some 4 mil black plastic and securing it with landscape staples over the area. If you are leaving it there for a month or more, the heat and lack of sunlight should do the trick. I am actually using this on my garden beds prior to planting. http://bartoy.blogspot.com/2008/03/dirty-dispatches-saying-sayonara-to.html Here is a link that might be useful: Life has taught us ...

The easiest way to ID weedy grasses is from the inflorescence, the seed producing stalk. Compare your weed to Dallisgrass, you could also try asking on the grasses forum or Name That Plant. Good luck!

The growth at the base of the tree is suckers from the rootstock. Ie they are part of the tree so you can't use herbicide on them or dig them out. All you can do is cut them off and try to avoid damaging the base of the tree as that can stimulate more suckering. Your other plant could be a smartweed but it's easier to id plants when they are still growing. W e also can't see the tops of the leaves.