I am always logged in on the devices that I run but as they are never switched on at the same time I won't have run across any of the problems you think might result.
However, I don't think that there should be a problem as, although each device is left logged in, this is only a state which is recorded by a cookie on each device - there is no continuous back-and-forth communication saying "here I am". Each time a device, which thinks it's logged in, accesses something on the forum then communication is set up and the forum software compares its database to see if the member is truly logged in and throws them off if they aren't or the login has expired.
The staff logged in is just a record of activity within a certain (probably settable) period. If you just look at a page without refreshing (lurking) then no contact has been made and you won't show up as being online.