You won't be able to remove  the New button  and leave the dropdown. It's
a single button. However, clicking the new button simply chooses the
default content  type for the library. You can select which content type
is the default  in the library settings. You can even remove the
default document content type  from the library and only leave custom
content types behind.

But the whole point is that I don't have a default content  type
(SharePoint makes me choose), but I want to have the user  choose the
appropriate content type. I have three content types and the user
needs to first select the appropriate one first, then add
information.

Your only real solution then is to train the users. If they click  on
the New button  instead of the dropdown  it will default  to whatever
content type  is labeled as the default content  type. I don't believe
there is a way to change this since it is an integral part of the
design.

I suspect that with a few lines of JavaScript you could change the
behavior of the New button  to do the same thing that the dropdown  does.
I don't have the bandwidth to write the JavaScript but I'm certain that
it could work.

What puzzles me though, is that if you create an item  and then edit,
you can change the content  type with a drop  down box. I would
really love for that box to appear on the new form. Just don't
understand why it appears on the edit, but not on the new.

If I only had a nickel for every "why" question...

Imagine a dime for every "how" question...

And then imagine a whole dollar for every "solution" answer!

Anybody have a work around or a get me by idea?
I'll let you publish it as your own thought and concept, I just want
to use it

The choice  appears on Edit to allow you to change a Content Type after
the item  is created. New assumes you are using the drop  down so based
on the app design it would be considered redundant. If you require
additional metadata on the library, you might be able to add  the Content
Type selection to that form, but I've not tried it myself, and you'd
operate under the normal restrictions for additional metadata (e.g. no
multiple uploads, etc).

The answer is that Microsoft designed the selection of the Content Type
to be on the toolbar not on the form for a new ListItem but there is no
workable toolbar interface in the context of editing a ListItem so
changing the Content Type is within the form itself.

Now, if you can tell me WHY they made that decision, you won't owe me a
dollar.