About to open my first wallet a couple of question before I have my own wallet.

Does having a wallet require personally identifiable information?  Do I need to use my real first and last name or company name, etc?

It seems that individual transactions in your wallet are always public. What about your total balance?  Is it possible to be a bitcoin millionaire without others knowing about it?  If you trade a few bitcoins for real currency every once in a while (i.e. make "withdrawals" for the lack of a better word), will people know that also?
Some online wallet services might require your id, however, if you create a wallet on your local computer there is no personal information required at all.

The balance of each address is public knowledge at all times, but neither is the owner of an address,  nor the fact that different addresses belong to the same wallet.

Some of this information can be gained by analysis of payment patterns or after information about ownership was revealed publicly. For example, addresses can be deduced to probably belonging to the same wallet, if both of them are used in the same transaction as inputs.

If you use big exchanges with KYC policies to trade your bitcoins for Fiat,  they will be able to tell on you. You might be able to make local exchanges for privacy.

In conclusion,  with sufficient know-how and micromanagement you can probably keep your total balance secret.