New iPhone, can't access prior iTunes purchases

Problem adding new iPhone to pre-existing Apple account, can't access music previously purchased via iTunes.

I have used iTunes on Windows-based laptops for years, with a very old iPod (3rd Gen?). Point being, I have had a working Apple ID for years. Most of my iTunes Library consists of CDs I own that I have ripped using iTunes. I have purchased a much smaller number of songs (several dozen) via the iTunes Store, and I am having trouble only with this music that I purchased from the iTunes Store (the CDs I ripped all sync fine).

I recently (last month) purchased my first iPhone, a 6s, iOS 9.1. During setup, it would not log in to my existing Apple ID, I had to create a new password, using the *same* email as my existing Apple ID, and my Apple ID (with new password) are working fine for all other Apple ID purposes, just not for music purchased from iTunes.

I also have a new (several months old) laptop, running Win10 (fully updated), and I have just installed iTunes (latest version, first installation on this laptop). I had never installed iTunes, or logged into the iTunes Store, or synced my old iPod, using this new laptop. What works fine: on the laptop, I can access my entire pre-existing iTunes library (stored on external HD). So far, so good. I have also synced all of my ripped CD tracks to my new iPhone 6s. Also good. So we know that iTunes is installed and working properly on Win10 machine, and that iTunes library is fine, and that new iPhone is syncing properly for the most part.

Problem: several dozen songs that I purchased via iTunes previously will not sync to my new iPhone. I paid for them, they downloaded and played fine on old laptop and very old iPod, and they appear and play fine within iTunes on new laptop. They just won't sync to new iPhone. I am using the same Apple ID (as far as I know, I certainly haven't changed my email address). The only things that I am aware are different: 1. new iPhone, 2. new laptop, and 3. new Apple password. (Yes, I did search, and I understand that I can't *merge* 2 separate Apple ID accounts; I don't think that's my problem. I don't have a separate login to go back and log in to the old account, the old password no longer works.)

So: how to get iTunes to let me sync these tracks? Any suggestions would be greatly appreciated. Thanks.