For question 1: try to come up with a proof that the language is not regular.
Hint: Let your language be $L$ and suppose towards a contradiction that it is regular. Use the closure properties of regular languages (in particular, regular languages are closed under intersection) to show that there is some regular language $L'$ such that $L \cap L'$ is a well-known non regular language.
Question 2 is not asking you whether all words in $\{0,1\}^*$ contain an even number of $0$s and an odd number of $1$s (that's clearly false). Rather, it wants you to consider the language defined as the set of all possible words (i.e., words from $\{0,1\}^*$) that also satisfy some additional property. This additional property is having both an even number of $0$s and an odd number of $1$s.
Hint: Try to design a DFA $D$ for this language. Suppose you are reading a word from left-to-right and you want to determine whether it satisfies the above property. What do you need to remember? Encode this information in the states of $D$.