Coding a custom control like DateTimePicker

I'll try to post some information which hopefully will help you get started, even though it does not directly address all of your concerns. First of all, you can get a free dotpeek decompiler, do CTRL+T, find DateTimePicker type and learn how it works. Be prepared to see a lot of winapi calls. Don't go this route unless you are confident that you will be able to comprehend everything and re-write it to work for your needs.

A more simple way would be to create a custom control. It may not look as nice, but at least you can get it working much faster. For example, create a UserControl1, which consists of a TextBox + a Button:

Public Class UserControl1
  Private Sub Button1_Click(sender As System.Object, e As System.EventArgs) Handles Button1.Click
    Dim frm As New Form2
    Dim p = frm.PointToScreen(New Point(Me.Left, Me.Bottom))
    frm.Location = New Point(Me.ParentForm.Left, Me.ParentForm.Top + 40) + p
    frm.Show()
  End Sub
End Class

Disclaimer: Doing +40 is a poor coding standard, but that's the best I could figure after struggling with conversion of coordinates to screen/client. If you find a better way - please suggest in comments.

Anyway, at design time it looks like this:

Form2 is your popup, it will close if focus is lost, to imitate real popup behavior:

Public Class Form2
  Private Sub Form2_Deactivate(sender As Object, e As System.EventArgs) Handles Me.Deactivate
    Me.Close()
  End Sub
End Class

At design time it looks like this (FormBorderStyle = None):

Compile and drop this control onto your main form. Run your application - when you click the button, ft/in dropdown will appear underneath your control. Just as any popup would do (including that of DateTimePicker), it will disappear if you click anywhere else (Deactivate event is used to make it work like this). Here is a picture, to illustrate its behavior:

Now add validation, formatting etc.