The third boss is definitely one of the hardest bosses in the game, the only other one that compares is the one from level 6. If you're really fast with the hatchets you can take him down as or immediately after he makes the second stack of blocks appear; to do this you want to jump towards him at the beginning of the fight and land somewhere that you can hit him (basically where the fist stack of blocks will appear), try to make sure that you hit him with 3 hatchets per jump (high jumps, hold II as you jump and then try and hit II as many times as you can while you're in the air) and cut it close with moving out of the way of the first thing he throws and then position yourself so you won't get hit by the second stack when it appears and keep hitting him with the hatchets.
A slightly easier but slower method is to get to the leftmost portion of the screen after he makes the second stack appear. Hit all of the blocks you can before they start falling (bottom 4) and then high jump immediately after the top 2 that you can't hit start to fall and take them out with the hatchets, after that hit the bottom 3 blocks from the 3rd stack, run under it, and start hitting him again as he walks back to the right side of the screen.
As for the coyotes coming from the left, remember that on 3-3 both will jump over you, don't try and jump over them or you will get hit. All of the others are ones you have to jump over but you'll always know they're coming because you'll pass a flower right before they start running after you. Jump over them and then try and land the jump immediately behind them so you can kill them but remember that they will come to a dead stop after you hit them one time, so don't get too close or you'll run into them before you can hit them a second time.
I hope this helps but unfortunately I can't seem to type this out as clearly as I can see it in my head. Unless you're really quick the second method is probably the one to go for, once you get it once you'll probably have the timing down from then on though. Maybe I'll see if I can make a video of the 3rd boss later on.
one clarification: sometimes I say "high jump" and other times "jump" but they're all high jumps; I'm always holding down II when I jump in this game