IPhone – Are all Verizon iPhone 5S are GSM Unlocked globally

Your logic is not correct. For a while, there were people offering unlocking services for around US$170 who are believed to have bribed an employee of Apple or a mobile phone operator to unlock phones using the official channels, and those phones also didn't need to be jailbroken. There's no way I know of to distinguish one of those phones from a factory unlocked one.

Assuming it is a genuine factory unlock, there should be no problems -- I have a UK factory unlocked iPhone 4 and have upgraded the OS several times, though I've never tried to do so in another country.

Will the iphone get 4g speeds

No it won't. iPhones, even 4 and 4S, are not 4G devices. The 4 is for the iPhone fourth generation, but they are in reality 3G devices. So you won't get better speed than 3G (for sure).

I'm not sure for the network itself since I'm in Canada and I don't look at US providers very often. From memory T-Mobile has a different (AWS) network than AT&T so that could also affect your connectivity and speed. You better check the phone versus provider specifications (or someone else doing the same as you).