Hi,
I just joined. I'm on t-mobile, pre-paid, but with the GS4 I can switch to 4G on a daily basis. I've had this new phone for about 2 weeks and just returned from a trip to southern California, where my car broke down and I got a very good chance to see how well the phone would perform (being on the phone frequently and for long calls)
It does get a bit warm, but I'm not sure if that's the battery or CPU. Location suggests it's the battery discharge generating the heat.
I switched it to Power Saving mode, but hadn't many long calls after the initial incident with insurance and various garages (insurer kept trying to shift repair from dealership to local garages, not that I have anything against them, I expect they don't keep the parts in stock and the delay would have been much longer.)
As for lag, I found the phone to be very responsive compared to the Gingerbread model I have for work.
Camera is rather wonky, seems too easily it decides to do burst mode, as I have a 32GB SD it's not really an issue) Does seem to be a bit frustrating to get photos at some times, which require the reticle to give you the OK, like my Canon 6D, geez how I miss stupid old 35mm film cameras which just took the blasted picture. Haven't found a better camera anyway, operationalwise.
Battery life seems short. There are 4 cores chewing through java, javascript, HTML5 and whatever, which is all bloat as far as I'm concerned. Web response time was pretty good, but really don't like the Google Maps interface. Keep hitting the icons on maps and it just moves it around like some April Fools thing from years go. Some tuning needed there.
Otherwise, it's a good phone as long as I don't have super high expectations - it's a communications device, nothing more.