It is fine for them to have a whole carrot to play with.

I have been advised to do this for my puppy!
She can play and snack on it at the same time...

The only drawback is its a bit messy thats all.

Yes, yes, yes, good for the teeth and the anal glands! Mine eat a big bagful in a couple of days!

Hollie loves raw carrots, she chews them really nicely so her teeth get the benefit, but oh boy she stinks afterwards!! The farts are horribly smelly!!:smt078

If you choose weight bearing bones they will be fine, can't understandy why marrow bones are a 'no' mine love them and are occupied for hours, and stops them tearing the pebble dashing from around my back door!!!!
Give them bananas and melon with the skin/rind still on as he will eat the whole lot.
Raw chicken wings and legs are fine defrosted, just don't cook them. If you feed tuna or any tinned fish choose the stuff in tomato sauce or sunflower oil, if you can only get it in brine, rinse the fish first due to the high salt content of brine.
Also your butcher may make his own pet meat, worth asking about as it is very cheap, also ask for chicken carcasses, necks etc.

My Ozzie adores carrotts. Keeps him entertained for a few mins. I keep them in the bottom of fridge and he goes loopy when I open the bottom drawer that they're in. So much better for him than most of the treats you can buy!