You appreciate and value everything your superior says?
I really appreciate how you have helped me with this project.I really appreciate your helping me with this project.
"Roger and I appreciate what you did" is correct grammar
The heart is superior to the liver
Your sentence is almost correct. It should be: "I appreciate the time you took from your busy schedule."
Both "I appreciate you responding to my letter" and "I appreciate your responding to my letter" are correct. The first sentence uses "you" as the direct object of the verb "appreciate," while the second sentence uses "your" to indicate possession of the gerund "responding." Ultimately, both versions are grammatically acceptable.
The knee is superior to the ankle
No
You had not taken that into account. Would be the grammatically correct version of that sentence.
No. Since "circumspect" is an adjective, "I appreciate your circumspect" is about as grammatically correct as "I appreciate your clean". Saying "I appreciate your circumspect behavior" is correct, similar to "I appreciate your clean appearance".
The correct punctuation for the sentence is: "How was your weekend?" Jerry asked.
This can fall under two different forms of sentence. One denoting a list, and one not. IE: Thomas is following Susanne. The following is a list of automobile parts you will need: So, this will largely depend on the kind of sentence you are forming and the information you are trying to get across to your audience.
The correct grammar is "Between you and me."