Thanks. There are so many questions (!)
Yes there are. I'll try to get through them one at a time.
In, say, wallets (such as bitcoin-qt or armory), when a bitcoin balance exists, what is the essence of the balance value
The "balance" is the sum of the values of all unspent outputs in the blockchain for which the wallet has the associated private keys. Note: the values are stored in the public blockchain, not the wallet. This means the value is "backed-up" in thousands of copies all over the globe on every single full peer.
- is that what the 'private keys' are? Is the 'value' associated with these private keys, like, they are the actual (money)?
The private keys can be thought of as special "passwords" that give you permission to "spend" an unspent output in the blockchain by creating a transaction that assigns the value to one or more addresses, and providing the public key and a cryptographic signature created with the private key. Once you've created the transaction, the outputs become "spent", and the new outputs can only be spent with the private keys that are associated with each of the new addresses. It is impossible to know what the private key is if you only know the address and public key, but it is quite easy for a computer to validate that a "signature" has been provided by the private key as long as you have the public key and address.
Am I correct in thinking that with, say, a paper wallet, it is the private keys which are printed out?
Exactly.
Is it unusual for the private keys to be shown visibly in wallets? I guess the answer may be yes unusual, because I hear talk of them but I do not see them offered as examples nor printed out...
With the exception of a paper wallet, the private keys are generally not shown. Other than make copies of them, there isn't much use to seeing the private key.
Am I correct in thinking that part of the usefulness of a wallet is to handle and convert private keys into a total 'value', regardless of the complexity of the composition of the grand total?
Yes. The wallet takes care of doing all the necessary math to figure out what the associated bitcoin address is for each private key that the wallet knows about. It then scans the blockchain for each unspent output sent to those addresses and totals up the associated value.
And sort of associated question - is it useful to try to arrange a transfer into my wallet of a single sum (single private key??) or try to split the value into more than one single transfer (several private keys??) for future convenience? Or is this somehow missing a point?
tia
If you break up the total into parts that are too small (less than 0.01 BTC), you can start to run into situations where transaction fees become necessary and even large. On the other hand you increase your privacy a bit by splitting up your balance. For most people it really doesn't matter much which way you do it.
