The farmer doesn't affect digging, and the alchemist is actually the only one who grows anything (herbs in inventory). You can 'farm' food in the game-sense when monsters respawn, which happens occasionally based on plot progression, since those include a large number of animals that provide meat.
The farmer's gimmick of eating food rather than killing monsters means that he levels up more quickly than most other characters, although this is offset by being initially weaker on the skill side. They're not super-powered or particularly weak, just different.
This really depends on how you're playing the game. If you avoid combat to progress through the game more quickly or because you find certain fights simply too difficult, a farmer may level a bit faster. But if you kill almost everything you come across, your other characters will level considerably faster. By the end of the game, my farmer was 2 levels behind my second slowest character (bear in mind that experience required to level increases exponentially.. he was level 12 with 93k xp whereas my level 14 rat alchemist had 197k xp). And I had been putting the +25% experience amulet on him every time I fed him for about 2/3rds of the game.
Something that might make it possible for a farmer to level faster would be to feed him any excess food you find, regardless of the nutrition limits. However, doing that might really hurt you in the long run. Towards the end of the game I was struggling to find enough food to keep my party fed. There are a few dungeons of considerable size that provide you with hardly enough food to survive. And if you take your time to search for secrets, you'll be using up a lot (and I didn't even rest much). I'd pretty much caught all the fish in the river and regularly had to resort to hunting in areas that provided 1-3 respawns of enemies that had a chance of dropping some food items (turtles, wargs, toads and giant snakes, mostly). And it seemed to take progressively longer for them to come back after each time I killed them.
Farmers might make a MUCH better party member in custom dungeons with ample amounts of food, but I found that having the class really didn't help me in any way compared to the other 3 characters in my party. What the class does accomplish is that it adds an interesting optional challenge to the game. Imagine playing the game with 4 farmers on ironman mode with single-use shrines. You'd have to be really amazing to be successful that way and it could be an interesting Let's Play idea..