Is there a max skill level?

The max level you can obtain permanently is 10, but food buffs can push you higher. You can see your effective skill level on the skills screen.

That said, you do NOT appear to gain efficiency from levels above 10. Casting an Iridium rod costs 7 stamina anywhere between 10 and 13.

At the very least, fishing above 10 makes fish easier to catch. When I was trying to catch Legend, I spent two full in game days (I quit, rather than slept, so it was actually the same day, twice) trying to catch him without ever getting him above 75%, and only above 50% once. The third cycle, I ate a dish of the sea (+3 fishing) and caught him nearly effortlessly the first try. I quit again, tried to catch him without the food buff, failed hard, ate the food, caught him easily. Either there's a placebo effect or food buffs work above 10.

Yes, the max skill level across all skills (currently) is 10. Once you reach level 10 in a skill you can still level other skills as normal. However, you can not surpass level 10.

Source: http://stardewvalleywiki.com/Skills