Can I make Bitcoin transactions without the involvement of a bank account whatso...

I am a web designer and I don't have access to a bank-account (and therefore, credit/debit-card) and I am not old enough to open a bank-account yet. Could I use a Bitcoin wallet to receive money and store it there as long as I want (and possibly, send those Bitcoins) ? Paypal would be ok except they only allow you to receive a limited amount before you are required to connect your Paypal account to a bank-account.

Could I use a Bitcoin wallet to receive money and store it there as long as I want (and possibly, send those Bitcoins) ?

Yes. You can buy Bitcoin at Bitcoin ATM where is located some places. You can find a Bitcoin ATM here. Some ATMs allow you to buy and sell coins, but some does not allow you to sell.