If you want to move your purchased programs that have been installed and registered with their manufacturers, in some (NOT ALL) cases you can uninstall a program from the old computer, let that uninstallation notify the software manufacturer over the internet during the uninstall process, then use the program's installer to install to the new computer using the original installation key. Whether or not this will work depends on the specific program.
I doubt that any of the OS migration programs you mentioned can handle that re-registering process automatically. AFAIK they just transfer data files between computers. They will move files from the old My Documents folder to the new My Documents folder, or from one Pictures folder to another, etc. That process does not install programs. Most Windows programs must be installed before they will run.
The safest option is to install new programs to the Win 10 laptop, and transfer your data files by copying them to your external drive from your old computer, then copying those to the new computer. It's a PITA but it works.
I am not sure if you forgot the sarcasm tag? You can download the Windows 7 installation disc ISO image file from Microsoft for free.
https://www.microsoft.com/en-us/software-download/windows7
Then you can purchase an “official Windows 7 key” from numerous sellers with good feedback on eBay for just a few dollars. I know absolutely nothing about the following seller, but here is an example link.
I have purchased a dozen or more licenses for various Windows versions over the years through eBay to use in virtual machines. I do not know the source of the licenses or their actual origins, but I have never had any difficulties with them if they were purchased from a seller with a lot of good feedback.
You can use these licenses in virtual machine, but it is almost a complete waste of time to try and get Windows 7 to work properly in a clean install on a new laptop with a 7th or 8th generation Intel core processor. It doesn't matter whether you want to use a 32 or 64 bit version. This is because Intel has not yet and may never release the drivers necessary. I do not know which AMD processors have the necessary drivers.
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I retract my “but you can do it” in post #19. You probably cannot do it if your laptop has a new generation CPU.