How to Install Old App Version onto a New iPhone with new iOS?

The answers will not help, but at least you will know that they are accurate, so you won't have to waste your time trying to find solutions that do not exist.

How to i install the old version that does not run iOS 16 or above on the new iPhone with an iOS that is 16 or above?

You can't.

Is there a possibility to spoof the AppStore that iOS 15 is installed (even if in fact it is newer) as it can be done with user agent in web browsers?

No

Or is it possible to downgrade and reinstall latest iOS 15 to install the app and update again to 18?

No

If the app developer of those older applications has not made current versions compatible with iOS 18+, then you can not install the older applications on it, nor spoof the App Store to believe that iOS 18 is an older iOS version.

I'd try contacting those apps developer/support teams and ask them for a possible solution (if available).

Use these instructions:--> How to contact an app developer - Apple Support

Regards

Giulio

Aipi wrote:

@Giulio: I already contacted both of the developers – they are both not interested in respecting the wishes of the long time users ...

In that case, I'm sorry but I believe you have no other options.

It's a choice of the app developers and, as you surely know, this doesn't only happen in the iOS "world", but also for devices that use different operating systems (Android, Windows, etc.).

Giulio

Aipi wrote:

What about transfer with third part tools (like anytrans etc.)?

Don't jailbreak your phone in an attempt to achieve the impossible.