iPhone - locked to a particular carrier (like Verizon) - is just a record on APPLE server. So if you put in your SIM card and iPhone does not complain (need WiFi connection to APPLE server) - it is unlocked and good to go.
Regarding LTE-4G - Verizon IPhone 5s - support a lot of bands (LTE frequencies) - but not all of them. People complain that 2600 frequency - used in Europe is not supported by any of USA models.

iPhone 5S A1533 LTE 700/800/850/900/1800/1900/2100 USA AT&T, T-Mobile, Verizon
iPhone 5S A1453 LTE 700/800/850/900/1800/1900/2100 USA Sprint, United Wireless
iPhone 5S A1457 LTE 850/900/1800/1900/2100 Europe/Middle East
iPhone 5S A1530 LTE 800/850/900/1800/1900/2100/2600 Asia Pacific
iPhone 5S A1518 LTE 800/850/900/1800/1900/2100/2300/2600 China