The biggest damage to those buildings is rain splash from the ground back up. Setting them on stones or other surface to keep the mud splash away is the thing to do. 

Do what we do for chicken pens-  get a roll of 24" hardware cloth, roll it out flat, put a bend in it in the middle.  Dig away the top 8" of dirt out 12" from the building, lay that L of hardware cloth in with flat side in the dirt, vertical side against the building.  Take a 2x4, and screw it to the building to sandwich the vertical side, repeat for the other three sides, overlapping the corners on both the horizontal and vertical planes, then backfill.

Use hardware cloth to cover vents.

The vertical portion will prevent critters from going under the exposed bottom edge.  The horizontal part will keep critters from digging down, then going under.  They'd have to step back a foot from the building, and dig down beneath it, to get in.

Mice will climb right up the sides, and chew under the soffits and gables, or through the bottom edges of the doors.  Thin sheets of galanized steel are your friend here-  bend it to wrap around the edges and corners, so they can't chew on them... then make sure the door closures pull in tight so that there's no gap they can wiggle through.

If you use poison baits, only place them INSIDE the shed, and inside screwed-down enclosures that your patrol cats cannot get to, or move... so your hunting-staff doesn't get exposed to it... and don't let your guardians get to the rodents which have succomed to the bait.

Keep grass seed, bird seed, etc., inside tin cans with tight-fitting lids... galvanized garbage cans, fancy cookie tins, the metal discs that those awful fruitcake came in, and of course, the tall metal cans that show up at the office full of three kinds of popcorn at christmastime work great... keep them on the floor, or on secure shelves where they can't get knocked to the floor and spilled open.