Weed #1 is certainly a type of Pondweed, probably American...I don't know my pondweed that well.
Weed #2 looks like Nutgrass to me. It is a grass that tends to grow easily anywhere the soil stays damp. Look it up and see what you think.
I believe you are exactly right...#2 is nutsedge weed.
Any advice on what to do going forward? Should I try to get rid of either? Leave it alone and just trim down as desired?
The Nutgrass should stay at the water's edge or on the wet soils nearby unless the water rises above it, but it won't do well partially submerged in standing water for long. I love the way it looks especially with seed. The only folks that don't like it are those that prefer a manicured weedless lawn.
The American Pondweed is deemed to be one of the most desirable plants one can have in a pond (per my PB readings)unless you don't like floating leaf/submerged plants OR your pond's conditions are conducive enough that it takes over too much of the surface due to shallow waters and fertile bottoms. I don't think many consider it aggressive. I am trying to start it in my pond. If it gets out of hand...I will go swimming and pull it out. It up roots pretty easy.
