do apps transfer to new iPad during quickstart?

Strictly speaking - no… but this does require explanation.

While associated App data is copied from device to the other during the transfer process, Apps themselves are not directly transferred between devices - either when using QuickStart, or when restoring from a backup.

Instead, a record of the installed Apps is transferred from the source to the target device. For Apps that are both compatible with the target device and remain available within the App App Store, the transfer process will automatically download and install the Apps on the target device.

It follows that for older Apps that have been withdrawn and are longer available, you will not be able to use them with the new device.

These support pages should get you started with setup of your new iPad - and the methods of transferring your data from old to new iPad:

Set up your iPhone, iPad or iPod touch – Apple Support

Transfer data from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support

Use Quick Start to transfer data to a new iPhone, iPad or iPod touch – Apple Support

If the donor iPad, from which you are transferring your data precedes iOS12.4, QuickStart will not be available.

“Transfer” is rather a misnomer. In reality, your data and setup is merely cloned to the new device. Nothing is moved or removed from the original device.

An alternative is to restore either an iCloud or iTunes backup to the new device:

How to back up your iPhone, iPad and iPod touch – Apple Support

Restore your iPhone, iPad, or iPod touch from a backup - Apple Support

Yes.

See the following Apple Support article for information on apps and other data transferring with Quick Start: Use Quick Start to transfer data to a new iPhone or iPad - Apple Support

Not specifically. No backup or transfer includes the actual app. If there's any restore or transfer, it may be of the supplemental app data, but the apps themselves have to be downloaded from the App Store. But theoretically it should be the same after it's done.

There was a time when iOS backups (including for iPads at the time) included storing the actual app in the backup. It was also possible to do app updates via a Mac or PC, but that could get really messy with apps updated via computer still being listed as available apps for update through the App Store on an iPad. But I digress.

There can be issues if an app is no longer available on the App Store. But it depends. Some apps are completely removed from the App Store without any way to retrieve them. I have an app that hasn't been available from the App Store for new users since maybe 2016. But when I've transferred my setup to new iPads I've been able to restore the app since the icon is still there on my setup. Even when it has offloaded the app, I've been able to restore it.