Must I have an account with every gateway I use as an issuer? In the Ripple Client, in order to convert to BTC, I need to select a Base Currency (it currently says BTC ???.) 

I searched for a Base Currency Issuer address and found how to grant trust to BitStamp, which lists what is supposedly BitStamp's address.  I realize, however, that I cannot trust this address because it comes from http://www.nofiatcoin.com/ instead of https://www.bitstamp.net/. 

I assume BitStamp requires signing up for an account before showing an address.

Is having an account just BitStamp's policy? Say I had a secure way of getting BitStamp's address without signing up for an account and I added this address as a BTC issuer. Couldn't I just buy/sell BitStamp-IOU's over the Ripple Network, even without an account at BitStamp? If so, why would BitStamp only divulge its issuer address if the user has an account?
Where can I find the address?

The address you need to trust to hold Bitstamp issued currencies is available from the Bitstamp website, but as you noted, you must be logged in to see the relevant page.

You can also find the address at https://www.bitstamp.net/ripple.txt under the [accounts] key. The ripple.txt file is documented here: https://ripple.com/wiki/Ripple.txt

Do I need an account?

You do not need to have an account with Bitsamp to be able to hold Bitstamp-issued assets. All you need is to trust their address. You can then buy Bitstamp-issued BTC, or USD or EUR or whatever else purely on the Ripple network.

Of course, having a Bitstamp account will allow you to redeem those IOUs through Bitstamp. Without such an account, you can still trade them on the Ripple network for either XRP or other assets, likely issued by another gateway.

You may find this primer to be of help: https://ripple.com/guide-to-ripple-gateways/