I created a paper wallet from coinbase with some initial amount and secured it. I want to keep adding more bitcoins to the same address. Can I do that? Are there any issues doing it?
Yes, you can.
The wallet itself does not hold any money. It is just a private key, to which the coins will be associated.
Yes. I have done so myself. I gave my kids paper wallets with 10 millibits each and am now giving them a bitcoin allowance of 4 millibits per month.