I am working on a Trek 800 21 speed. I am having an issue with the rear derailleur. It won't shift down to 7th gear. If I manually move the chain to 7 it stays there, and shifts back up normally. But when I try to shift back down it immediately refuses to go into 2nd until it is shifted to 3rd. I can move it back up to second and it will stay in 2nd gear but repeats the refusal to move again.

Replace the cable and housing. That's the only advice I can give w/o seeing the bike.

https://www.parktool.com/blog/repair...eur-adjustment

Best throw something at it and hope it works is the cable and housing mentioned above, its a semi-regular maintenance item and worth trying. Before trying check the hanger.
Stand behind the bike with it straight up and down and look at the der hanger, the part the rear der bolts to, is it straight up and down. If you can't tell easily either way by looking but the chain in the middle ring in the front and the largest gear in the back and see if the pulley cage is running straight up and down, if it looks bent you may need the hanger straightened. It is a steel frame so this is less likely but can still easily happen and is often shown by the bike not able to drop into the smallest gear as dropping the bike or crashing it often bends the der towards the center. You will need the tool to do the job, a skilled mechanic can eyeball it fairly well on a 7 or 8 speed bike like the trek 800 but it requires the tool to get it best.

Here is a couple of pics. Sorry the one is so blurry.

Sure sounds like cable friction is greater then the rear der's return spring strength. This is one problem with Grip Shifter compared to trigger shifters. The grip shifter has no helper spring that adds some cable push at the bar end of the cable system. Then the low grade level of the rear der (pivot slop, weak return spring and often too much open chain between the cogs and upper pulley), chain wear and cable sliding condition all add their aspects too. The advise to replace cable and casing is a good one. Check the chain wear. Unfortunately most people only understand the chain's lengthwise wear but it's the chain's side to side flexibility that effects shifting (especially with front shifting). Andy

Would seem to have been a cable stretch issue. A little time seems to have been what was needed.

In the bottom pic, what gear do you call that?

The pic is blurry, but it looks like it is in the 5th position of a 7 speed.

On traditional top normal the RD return spring moves the chain to the small cog.

John

Always nice when the problem fixes itself

Cables don't stretch.
Maybe a bit of spray lube in the RDER pivots would have sped things up. It looks pretty dry/dusty/rusty.
OR
the old, stiff grease got "worked enough" to thin out a bit.
Wait until a noticeably colder day....

What is commonly called cable stretch is the casing and it's end caps bedding together. Ironic that a lengthening reference is actually a compressing situation. Andy