All of the carriers rolled out LTE to the metro areas first, slowly expanding to the rural areas (recouping your investment in technology quicker that way). All legacy GSM carriers phones will work in rural areas that may only have EDGE/HSPA/HSDPA support. CDMA carriers may also still have LTE "dead zones" that only have CDMA/EVDO support, but their phones will also work in the legacy GSM 2G/3G areas (if roaming is supported).
All cellular phones are designed to maintain your connection to heighten the user experience. A GSM-only phone is constantly comparing the strength of the signal from the various towers to determine optimum path. In Europe, the rules may be such that a number of carrier connections be allowed, causing the phone to roam often, but providing the best user experience possible.
Unless entirely disabled by the carrier a CDMA/GSM phone will still poll with the secondary radio - the polling interval could be long, causing trivial battery drain, but they could still have it not entirely disabled. (this is more and more likely to be entirely disabled over time as the number of CDMA carriers approaches 0). I'm sure AT&T would entirely disable the CDMA radio as they have a very rich GSM roaming partnership internationally....the others? Likely it'll be polling whenever you roam.
