Do I need two passwords? I keep getting an annoying request to enter my iTunes password, and I have never even tried to order anything from iTunes
You don't necessarily need two passwords/accounts. Here is how it works:

You have an iCloud account, and an iTunes account. Both are accessed using your AppleID account credentials, but they are separate services. So account changes made in one service (like change of password) will be applied to the core AppleID account, and by extension the account used to log in to all other related Apple services. Most people use the same AppleID account credentials for both services, but you can use two separate accounts if you prefer (I do this for managed business devices).

iCloud accounts are generally only used to backup data and configurations (Activation Lock and Find My iPhone are also features of iCloud). You typically never get prompted for this password unless you are using a service or App that explicitly wants access to your iCloud (usually to perform or access backups). The iCloud account credentials can be set at Settings/iCloud.

iTunes Store accounts are used to purchase, download and update Apps. Depending on your settings, Apps may try to update themselves in the background, and this may require you to verify with your iTunes account password (even free Apps). This also occurs with iOS updates, but with a 1st gen device you won't have to worry about that. This is most likely what is happening to you. The credentials for this account can be changed in Settings/Store on 1st generation iPads. You can also disable automatic updates for Apps via the same location. That should stop the prompts. I have no idea what could be trying to update on a 1st gen device, but I am super curious.

Hope this helps. Apple device/account management can be unnecessarily convoluted sometimes. I wish someone would fix that.

Good luck.