There is a slight difference. "Try to count sheep" suggests the person find out if they are able to count sheep. Maybe they are a child and are still learning how to count. Maybe they're so tired they may not be able to count well. "Try counting sheep" suggests the person find out if the act of counting sheep (which we assume the person ...
... based on studies [performed in our lab]
The structure is fine. "Performed" is not an adjective here but a past participle verb form.
"Performed in our lab" is a subordinate, past-participial clause modifying "studies". It has the same meaning as the relative clause in "based on studies [that were performed in our lab]&...
The only way to know is to ask your teacher. Your sentences are grammatically correct, so the reason for a bad score must be that you aren't doing what the teacher asked you to do. For example, perhaps she told you to only use second conditionals. Or perhaps this was just the first question out of ten.
But I can guess what "conditional chains" ...
Yes, it certainly can. However that sentence is actually ambiguous, because without context, which is more likely to refer to the more recent "having handshakes" than to the earlier "bowed a little". But grammatically it is fine.
You're quick to jump in as always.
This example is fine.
In this case, we are not using the as-as correlatve conjunction but the idiom as always.
Definition of as always
—used to say that something was expected because it always happens
As always, dinner was delicious.
"It is man ..., isn't it"...The statement uses the dummy pronoun, and the verb "is" so the tag follows.
But tag questions are part of conversational speech. They are a way for one person to check that the other person understands and agrees. But the sentence is formal and philosophical. Not something you would say while chatting.
"Being" has a few definitions, but it can be the present participle of the verb to be.
Saying "we think of the day as being the time from [x] to [y]" is a bit like saying "we think it to be...".
The reason it isn't quite the same as 'A equals B', as you put it, is because it is about perception. Your original statement is not ...