Yes it does. Both 4g and LTE
No this phone has 3G capabilities
Yes, this particular phone (model S960L) is true 4G LTE. However, it is on the Sprint network and we all know that Sprint's LTE network is not as extensive as the other major carriers. Sprint LTE is being expanded pretty quickly, though.
Be careful when ordering. There are two versions of Straight Talk SIIIs. The other version has the model number of S968C and uses the Verizon network but is limited to 3G. It is NOT 4G LTE. If you do not see "LTE" in the description of the phone, it is the Verizon version and will be 3G.
So, if you are in a good Sprint coverage area that has 4G LTE, this phone will be the ideal one to get as it will be blazing fast.
If you are not in good Sprint coverage, you may want to consider the S968C.
They are essentially the same phone except for the carrier and the LTE issue.
ETA: here is the Sprint native coverage map if you want to check your 4G LTE coverage. It may not be totally up to date as Sprint is rolling out LTE rapidly and they claim to have 4G LTE in all their entire native coverage area by the end of 2014 ...wishful thinking IMO. copy-and-paste: