Please, please do not sit in your base smacking a rock to train. You get far less XP than you would from killing enemies and it's boring. You can hunt, fight, craft, forage, gather, mine, chop, explore, fish, and sail at night. All much more fun than smacking a rock repeatedly to maybe gain a few skill points.

Using your axe for combat and for chopping means double the durability loss if you're running around doing both so many players have a different weapon for fighting. If you like using axes to fight, then by all means keep using it. Weapons can have different secondary attacks, different attack patterns, different speeds, and different special effects. A huge mallet isn't going to feel the same, or swing the same, as your axe or a sword. Also some weapons do more damage, or less damage, against certain enemies. Some enemies have weaknesses or resistances. Smack a skeleton with a club or mace instead of a spear or arrow, for example, and you'll see a big difference in damage.

Make the sword, fight with it a bit and see if you prefer that over the axe. I tend to prefer maces, they're very effective against many enemies and have a nice secondary attack, but when I make new characters I try to play using a different melee weapons for variety.