I have a 7 speed bike. The 5-7 gears don't move the derailleur. It stays at the smallest cog. The derailleur doesn't move until I shift to gear 4. When I get to gear 1, the derailleur is at the 3rd largest cog instead of the largest cog. Should I unscrew the screw circled in red in the first picture below to adjust the cable tension? If not, what other adjustments should I do? I've included additional pictures for reference.
Yes, you are on the correct track. There is too much slack in the cable so the shifter can't pull the derailleur inwards enough to engage the lower gears. To remove slack you change to highest gear (smallest sprocket), wind any barrel adjusters in, undo the cable pinch bolt (that you circled), pull the cable slack out and re-secure.
Before you do that though, look for the cause of the slack. Inspect the cable run from he shifter to the derailleur making sure the cable has not come out of a guide and that the housing is properly inserted into frame stops.
You'll need to adjust the derailleur indexing after fixing the cable slack, that includes setting the inner and outer limits and adjusting the indexing. Park Tool's article and video gives clear step by step instructions.
I'd also thoroughly clean the chain and sprockets and re-lube the chain, that will make shifting a bit easier.