What is a moderate Republican?

[bedstuy]

Some mix of the following qualities:

-Moderate on some social issues: pro-Choice, pro-gay marriage, not a fundamentalist Christian
-Believes in the basic government economic programs of Medicare and Social security
-Not racist (Not a birther, not afraid of Muslims, supports immigration reform).
-Not a tool of industry lobbyists particularly oil/gas, insurance/medicine and banking.

In my opinion, there are arguably zero moderate Republican members of Congress or in statewide elected office.

[bedstuy]

So... What issues are a moderate Republican allowed to be conservative on?

Assuming they aren't a True Leftist, probably fiscal ones. People who are moderate on everything but foreign policy or social issues, tend to get tagged with the "neocon" or "fundie" label.

I think you're just missing the point here.  Moderate isn't just being between the two empirical extremes.  The moderate position on slavery isn't "some slavery."   

Mark Warner is a moderate.  Mary Landrieu is a moderate.  If Mark Warner was a member of the Republican party, he would be a moderate Republican.  Once upon a time, you had Nelson Rockefellers and Harold Stassen in the Republican Party.  Now, you have only a handful of national Republicans that can lay claim to being moderate.  If you're truly moderate today, you don't have a place in the Republican party.