My question is not if I can upload two apps with the package names <TLD>.<name>.app and <TLD>.<name>.app2. (the app's name changes here)
It is if I can upload 2 apps with the package names <TLD>.<name>.app and <anotherTLD>.<name>.app on Google Play. (the top-level domain changes here, for example, one with "com" and another with "org")
Also, can 2 of my apps have different <name>s?
Will I have to have another Google Play Developer account for that?
Yes, you can publish two applications with different package names. So you can publish
<TLD>.<name>.appand<anotherTLD>.<name>.app. For Google play app they are treated as strings so they will be considered as different