Plural or sigular with collective noun

stmt1) the army does not commit mistakes.stmt2) the army does not commit a mistake. stmt3) the army never commits a mistake.stmt4) the army never commits mistake. stmt5) cow eats grass. ( here I want to say in a general sense whether a cow eats grass or not.) so should I use "the" before the word "cow"?

Q1) are these all sentences grammatically correct?

Q2) in stmt2, is it required to put "a" before the mistake word.