Not all BitCoin wallet's can alter the transaction fee - can they?
Any Bitcoin wallet can and must set a transaction fee. Most wallets allow the user to change what the wallet sets, but not all. However all wallets must still set a fee.
How low a fee can be set?
You can set as low or high a fee you want. You can set a fee of 0, but your transaction may take a while to confirm.
Is there a too high for that matter?
What do you mean by this? This does not make sense.