Do the traits have to be carried in order to be passed on to a Sylesti's offspring? Example: a pair of Generated Syleti with different traits (Say Mom has wings and runes and Dad has stripes and a lynx tail), but they were created as visible, not carried, is there still a chance the offspring could inherit one or more of those traits?

In your case of using generated 2-Vis parents, the offspring will inherit all of the traits but only the more dominant traits will show. The other traits you won't be able to see which is what the community calls a "hidden trait", or a trait that is recessive. If you scroll down to the end of post #2 here there's a small chart on visible breeding that might help.