When you turn an iPhone into an iPod touch
My iTunes ID and password are now associated with my new iPhone 5s. The old iPhone (an iPhone 5) still connects to my home Wi-Fi network, but in order to download new apps, I need to enter my iTunes ID and password on the old phone. I'm concerned that if I enter my iTunes ID and password on that phone, iTunes will consider it to be my primary device and at the same time disable my new phone from iTunes. Is this a danger, particularly when I've also associated that ID with an iPad and MacBook Pro
Not at all. Let me explain.
As Apple tells us, you can associate your Apple ID with up to 10 devices for use on Apple's stores (a maximum of five of these devices can be computers). After you've associated a device with your Apple ID, you can't use it with a different Apple ID for 90 days.
According to the iTunes Store's Terms and Conditions (which I'm sure you've read), an Apple TV is not an "Associated Device" and so doesn't count against your device limit.
So, if I'm counting correctly, you've associated your Apple ID with four devices--two iPhones, an iPad, and a MacBook Pro. Clearly, it's time for you to run out and buy more Apple gear!
Moving along, Apple's stores and iCloud service have no notion of a "primary device" in regard to iPhones, iPod touches, and iPads. When you replaced your iPhone 5 with a 5s, no switch was flipped indicating that the latest model would get something that was denied to the others--save for the ability to make and receive phone calls and use cellular data, of course. As far as Apple is concerned, your old iPhone is, for the most part, an iPod touch. Downloading apps to it won't signal Apple that you want it to replace your newer phone.
Bottom line: Enjoy a worry-free experience with both your new iPhone 5s and your fancy new(ish) iPod touch.