rear derailleur problems

If you undo the bolt securing the cable to the rear mech does it shift on to the smallest sprocket? If it doesn't then check the mech to see if it's free movement is obstructed by some gunge from the road and give it a good clean until it moves freely. You shouldn't need to lube a rear mech anywhere but check for signs of corrosion around the moving parts. Check the sections of bare cable for corrosion particular where the cable goes into the housing. If you find any rust then it will cause the cable to stick so replace the cable and the housing. If everything looks OK but it is still giving you trouble, the chances are that the cable is fraying at the very top, inside the lever. Replace the cable and housing. Always replace the cable and housing together for best performance, the rear cable always wears out before the front one. It's fiddly but easy to do and well worth buying your own housing and cable cutters.

Well without inspecting it I would guess that the first thing to try would be to tighten the cable by turning the barrel adjuster at the rear mech anti clockwise, try a quarter turn at a time until it shifts down fine. If it still shifts up just as well then great, you have adjusted your mech, if not you will need to replace the cable inner at least and probably the bottom piece of housing. Aside from that I would have to see it or take it to the bike shop, they will sort it out straight away because they have nothing better to do this time of year.

There are many videos on YouTube showing how to make the numerous possible adjustments to a rear mech. Some are very poor but search until you find a good 'un, it will be worth its weight in gold.

I'll second that. Problems shifting down usually means the cable is sticking. Probably time to strip snd clean or, even better, replace.

Have you tried pulling the cable through. They can stretch over time. Try shifting down onto the smallest cog then loosen the bolt securing the cable. Pull it taught then re-tighten the bolt. Also chains and derallieurs get clogged with muck at this time of year. So a good clean and lube might help. Hope this helps