Let me elaborate on a comment of bof. For positive integers $r$ and $s$, let $c(r,s)$ denote the smallest positive integer $n$ such that if we color the edges of a clique on $n$ vertices with red and blue, either the red subgraph contains a cycle of length $r$ or the blue subgraph contains a cycle of length $s$. The main theorem of Faudree and Schelp [All Ramsey numbers for cycles in graphs, *Discrete Math*. 8 (1974), 313-329] is the following.

(i) If $3\le s\le r$ and $s$ is odd and $(r,s)\ne (3,3)$, then $c(r,s)=2r-1$.

(ii) If $4\le s\le r$ and $s$ and $r$ are even and $(r,s)\ne (4,4)$, then $c(r,s)=r+\frac12s+1$.

(iii) If $4\le s < r$ and $s$ is even and $r$ is odd, then $c(r,s)=\max(r+\frac12s-1,2s-1)$.

So the answer to (1), to guarantee a monochromatic copy of $C_m$, comes by setting $r=s=m$. To complete the answer, we mention that $c(3,3)=c(4,4)=6$.

Discrete Math.8 (1974), 313–329 and earlier work referenced therein. $\endgroup$ – bof Aug 10 '17 at 9:20