Q: How do I know if the apps from my old iPhone will transfer to my new iPhone

You can also try to back up your iPhone to a laptop. Then use the laptop to transfer data to your new iPhone. For more information, visit this page: Transfer content from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support

No such Apps exist unless your iPhone is jailbroken.

Non-jailbroken iPhones can only get Apps from the App Store App or iTunes on your computer.

If you jailbroke your iPhone to install Apps from other sources, all bets are off, and no way to guarantee anything will transfer, and no support from Apple to do so either since Jailbreaking is not supported and results are unpredictable.

You should be able to see your backups in Settings->General->Storage & iCloud Usage->iCloud->Manage Storage->Backups.

You won't be able to see inside the backup, but it should contain data from all the Apps that were set to backup. Apps themselves will be re-downloaded from the App Store.

If you backed up to iTunes on a computer, Apps can be restored directly from iTunes.