On which machine can I install OS X Server? Macbook or MB Air or Macbook Pro?

Apple's Server.app will run on any Mac that can run Yosemite so all of those will work. However a laptop is a very unusual choice to use as a server unless this is purely for testing websites.

If your purely wanting to test websites then this maybe an alternative to look at.

https://www.mamp.info/en/

An MDM solution is one I would not consider to be suitable for a laptop. The MDM server needs to have a properly registered host name and IP address that is contactable all the time via the Internet so that the mobile devices can always find it.

A laptop almost by definition is not going to meet these requirements unless you were prepared to leave it plugged in at the same place all the time and treat it like a fixed desktop or server. For Apple's Profile Manager which is part of Server.app most people use Mac mini computers.

I will mainly use it as MDM for the IPads.

If you're just talking about a few iPads, you don't need Server for that. Use "Apple Configurator" from the App Store. It works over USB, not the network.

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Another option to look at which can be free is Cisco Meraki Systems Manager. See https://meraki.cisco.com/products/systems-manager

I use this myself. It does not do absolutely everything but it does do most things.

That's the link I found.