Does my customer get an email when I create an order for them from the admin?

Not by default. Orders created from Orders → Manage Orders → Create order don't send the confirmation email unless you ask for it.

Before clicking Place Order, tick Email order confirmation to customer in the payment section of the order panel. The customer then gets the same confirmation email as a storefront order, sent to the Email entered in Contact & shipping address.

This is off by default so phone and counter orders don't email people unexpectedly. If you already placed the order without ticking it, you can still contact the customer yourself.