If you buy an eagle AXS upgrade kit, it comes with the old rocker style shifter, a battery and charger, and a mech, obvs. If you want the new pod style shifter, you need to buy that separately. (It’s part of the transmission groupset, not eagle) Not sure there are Transmission upgrade kits officially, but some shops may sell you the minimum required kit (which may include the cassette and chain)
If you buy a mech separately, it doesn’t come with a battery, charger or shifter.
Eagle will work fine with shimano cassette and chain.