The achievements are all based on achieving level 10 in your skills. Is this the max level for a skill? And if so, when you reach this level can you continue to gain proficiency in equipment separately or are they directly tied to each other in a preset fashion?
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.