Is 'does' ever written as 'doe's'?

No, 'does' is never correctly written as 'doe's.' The word 'doe's' is valid, but only in a distinct context: it functions as the possessive form of the singular noun 'doe' (a female deer), referring to something owned by one female deer (e.g., 'the doe's fawn'). Importantly, 'doe's' is not a contraction for 'does not' (that contraction is 'doesn’t'). Additionally, 'does' as the plural of 'doe' does not use an apostrophe, and the auxiliary verb 'does' never requires an apostrophe (e.g., 'Does he play soccer?').