What is the proper way to get a new ID for a new record for a self-maintained primary key aka idkey?

Since you manage the ID yourself, in the general case it may not necessarily be a simple counter, but, for example, a Fibonacci sequence. If the source code of the class or table is available, it is better to see how the ID is generated in it. If there is no source code, and you are sure that this is a simple counter, then in my opinion it would be easier to make a standard ID and let IRIS manage it itself.

You can get the required Global reference like this programmatically:

 ; get compiled class  with your classname
USER>set classname="oex.Dir"
USER>set dic=##class(%Dictionary.CompiledClass).%OpenId(classname)
 ; get relationship to CompiledStorage
USER>set stor=dic.Storages.GetAt(1)
 ; get name of the ID-Global
USER>Write stor.IdLocation
^oex.DirD
USER>

to be clear: 
USEEXTENTSET is great for very large tables where the top (with the ID) is many 
storage levels away from data blocks.
It's up to the dimension of your project if the eventual gain in speed pays off at all.
for small tables as we know from our samples, I doubt. 

With a system-assigned %ID value (the IDKEY is defined as "system assigned" internally), you cannot override the value as there is no settable property. The way to work around that restriction is to define a property as the IDENTITY. It has similar behavior in that its value is system assigned but you can also override that assignment by specifying the value yourself. You have to set a switch that allows you to do that but it is possible. This type of behavior is similar to using sequences in PostgreSQL where some number of inserts that specify the id are followed by an update of the sequence used for id auto-assignment to ensure uniqueness later. Of course, when using IRIS and you specify the identity value, you need to be certain that a future increment (uses $sequence most likely) won't produce a value that will not be unique. 
This is a simple example of a class using identity.

class com.intersystems.App extends %Library.Persistent {

property id as %Library.BigInt [ identity ];
property name as %String;
classmethod add(name as %String) as %Integer {
    set me = ..%New()
    set me.name = name
    do me.%Save()
    return me.%Id()
}
}

I then implemented a simple unit test:

method TestAddWithId() {
    set obj = ##class(com.intersystems.App).%New()
    set obj.name = "E2"
    set obj.id = 100
    try {
        $$$ThrowOnError(obj.%Save())
        set id = obj.%Id()
        do $$$AssertEquals(100, id, "TestAdd() - add 'E2' with Id = 100")
    } catch exc {
        do $$$AssertTrue(0, "TestAddWithId() - Failed, exception caught: "_exc.AsSQLMessage())
    }
}

And the result of executing that test is:

The solution to this problem is to turn on IDENTITY_INSERT. I updated the test to do that.

method TestAddWithId() {
    set old = $system.SQL.SetIdentityInsert(1)
    try {
        do $system.SQL.Execute("delete from com_intersystems.App")
        set obj = ##class(com.intersystems.App).%New()
        set obj.name = "E2"
        set obj.id = 100
        $$$ThrowOnError(obj.%Save())
        set id = obj.%Id()
        do $$$AssertEquals(100, id, "TestAdd() - add 'E2' with Id = 100")
    } catch exc {
        do $$$AssertTrue(0, "TestAddWithId() - Failed, exception caught: "_exc.AsSQLMessage())
    }
    do $system.SQL.SetIdentityInsert(old)
}

and now the test passes: