What should I do with nothing to strengthen?
If I have nothing to strengthen in a language should I just strengthen skills with that button or strengthen skills I want to? Or should I just wait for skills to unstrengthen? Is there a "mental mechanism" I'm supposed to follow? Or can I just do whatever I want?
It's definitely not necessary to wait for skills to lose strength before they are all strengthened. There is no such thing as over-using the "Strengthen Skills" button! Many words that you have forgotten will ease back into your memory. Duolingo can make accurate guesses at which words are fading out of your memory, but sometimes it overestimates the power of your mind.
Strengthening skills that are already golden will lead to a longer time before they decay. In other words, you can prevent them from decaying by addressing the problem before it gets worse.
Also, you don't have to stick to strengthening. Learning a language should be fun! You can play around with Duolingo, Tinycards, whatever you want. Feel free to use an external resource - for example, now that I've finished the Spanish tree, I use this resource. I can't understand a lot of words, but it's immersing me in the Spanish language, and that is very valuable.
But, to answer your question, you can basically do whatever you want.
What if I get the opposite - having to strengthen skills every time? I almost find myself unable to just learn new skills without having to regild others, which bothers me.
That is a good question! Sometimes it can be very frustrating - you want to learn more in the tree, but you don't feel like it's okay to learn when your tree is not mostly golden.
I would recommend, just daily, before doing your "new skill", to do some "old skill lessons". In other words, do a couple practice lessons daily before learning new stuff. In addition, starting the strengthening from the bottom of the tree and going up can help you strengthen multiple skills in the same lesson, thus easing the amount of work necessary before learning is started.
Also, hitting the "Strengthen Skills" button when your tree is golden can lower the rate of decay.
Yes, using the Strengthen skills is a very good idea. Duolingo's algorithms don't actually directly keep track of how strong a skill is. They keep track of the strength of individual words. Strengthen skills will give you a lesson made up of weak words, even if the skills as a whole haven't weakened yet.
...and, as others have said, it might be time to branch out; as well as keeping your tree gold, you could find a children's book to read, listen to the radio, find a language exchange partner, etc.
Do you have the timed practices equipped? You could do timed skill strengthening,which is a bit more challenging. You buy the timed practice from your Lingot store (although with all the website updates I'm not sure if it's still available), and you only have to buy it once and it'll remain equipped and it applies to all the languages you're learning.
If you have a fully gold completed tree with nothing weakening, I'd guess you're probably ready to fly the nest. Try reading a YA novel in your language; watch a tv show without subtitles; go to lang-8 and write posts in your language. Duo will only take you so far. And as others have said, if you want to keep going on Duo then strengthening gold skills certainly won't do you any harm either.
If you have a concrete idea of where you're weak in a tree (you really should be able to know yourself better than Duolingo does), you'll probably do best to focus there — and that might well mean using external resources to shore up something that isn't quite clicking.