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[CriminologyProf]

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[shorena]

If I were to transfer my bitcoins to a paper wallet and then place that paper wallet in a safe deposit box, what are the chances that I will retrieve the paper wallet several years later to find that the bitcoins have been stolen?

If you created it properly on a clean offline system chances are effectively 0.

Should I regularly check the balance, and if so is there a way to do that without having to sweep and reload the wallet?

Doesnt do any harm to check. You can just look up the address in any blockchain explorer or use a watch only wallet.

Further, would using the private key to check the balance or sweep the wallet, does this jeopardize the security of the private key?

Yes, importing it to an online system to check the balance renders the idea of a paperwallet (offline) useless.

[NorrisK]

Regarding monitoring the balance:

You get a private key (secret) and a public key (not secret). The public key is safe to enter in a website such as blockchain, which will show the balance and any incoming and outgoing transactions.

Your HW wallet generator will show both the private and public key. It will also be very clear in which one to keep safe and which you can share.

[pawel7777]

So there is no need to ever use the private key unless removing or spending coins?

Correct

How would I check the balance using the public key?

Public key, in this context, means your bitcoin address (which is a hashed version of the public key).
If you use bitaddress.org to create paper wallet (or just to generate new address with private key etc) you'll see there are 2 parts: one marked as 'secret' - this is your private key (keep it safe and don't share this one), and the other, marked with green 'share' is your bitcoin address. You can use the latter to monitor your paper wallet balance.