Am I ok (license wise) to run two instances of Veeam One, one at each site, to make DR easier?
Do you want to monitor same infrastructure(set of objects) from 2 sites? If yes, you need 2 licenses. Why not to install Veeam ONE client on the 2nd site?
Will having two versions running add too much load on the vcenter, or would it just be best to run a cold standby?
In general it`s recommended to install Veeam ONE server on a dedicated server, however, if you have small infrastructure and/or really powerful hosts, you may try install them alongside.