The simple answer is, yes. The Xbox One has many features that require more bandwidth such as video and music streaming (as most content is served in higher resolution). However, these services are not required to game.
Services such as chat might require higher bandwidth but only marginally as Codec's, what is used to encode and decode the voice before sending it, allows for higher quality sound to be compressed into less data.
By your description I believe that you do not fully grasp the reason why you are lagging out. Indeed, lag is linked to bandwidth but it is usually not the deciding factor. What is more important is latency- Please check a site such as Speedtest.net to see what your latency is.
Latency is the round trip time of data you send out. Basically, imagine it as a postal service. Latency is the time required for a letter you send to be responded to, i.e, when you hold the reply in your hands and can read the response. This is a very important concept as online servers rely on the responsiveness on all participants. So when the host server detects that you take to long to respond to the packets it sends your way, it will drop your connection to avoid impact on the quality of the game for other players. That is why you lag out.
Therefore, a low latency is what you want. However, latency is only loosely tied to bandwidth. It has to do with the quality of your equipment, if you are on wireless or wired, your ISP, your geographical location, and the location of the remote server. There are more factors that are basically related to the above. So when you play on a server hosted on a different continent, your latency will be higher. That is normal and expected.
Now, sudden drops in latency are usually what cause sudden lag or drop-outs. They are caused by bursts of data that clog up the "pipe". Again, imagine it as a postal service. While on a normal day they can receive and forward your mail in a few hours, during Christmas season they are swamped and will take longer. This is to be expected. Similarly, if you are saturating your connection or equipment with downloads, streams, or similar, your equipment, especially older equipment, might struggle to forward all packets in time and connections might drop.
The tricky part is that there is not always something you can do. If you live in an apartment building that shares a physical line, the other tenants might have some activity that is transparent for you but still impacts your connection. The same might hold for your street as all connections on a street usually share the same physical line which uses the same physical equipment down the line.
One way to go might be to contact your ISP and ask what you can do. They should be able to help you. Personally, I'd advise you look into getting on a connection based on optical fibre. They feature less latency and can carry more bandwidth while alleviating some of the problems I mentioned above. If you want you can take a speed test and post it as a comment, I'll try to give you more streamlined advice.
First of all, don't worry; 50 degrees is perfectly normal.
For future reference, the temperature of the air surrounding your console's main outlet vent is not necessarily reflective of the temperature of the CPU or graphics card(the primary generators of heat in a game console). As far as I know, there are no diagnostic tools to show you more precise information on an Xbox One, but modern CPUs and graphics cards will usually reach temperatures of 70-80 degrees under a reasonable amount of load without damaging the hardware.
The other thing to note is that the top vent of and Xbox One is the outlet vent. There are also 3 inlet vents on the sides and rear panel. If any or all of these vents are blocked by anything, be it paper or the cases of video games, the whole system may be experiencing sub-optimal air flow, which would reduce the effectiveness of your Xbox's cooling system.
Also,the CPU and GPU both have embedded electronic thermometers which are constantly monitored by the Xbox itself. If temperatures ever get abnormally high on the actual hardware components, the Xbox will give you a warning with ample time to clear the air vents or turn off the console. If they ever get dangerously high, it will turn itself off immediately(I would estimate that dangerously high is greater than 80 degrees).
Happy gaming ;)
Best Answer
The Playstation 4 does not need an internet connection, however the day 1 patch does enable a wide variety of features. If all you want it for is to play games though, it should work entirely internet free I'm fairly sure
Games should come with the system update version they require to play, if any, a la the PS3 I believe, but I haven't encountered this yet since the PS4 updates it self...on the internet, anyway. You should also note a few games require day one patches either for stability or performance tweaks which could be a deal breaker. A couple games require patches for higher resolution, and BF4 at the least requires quite a few patches (but it's not much of an offline game anyway).
The Xbox One needs a patch to be used offline and will be basically dead with no connection or patch of any sort. Again, not sure if you can update the system's OS using game disks like you could with PS3, if someone knows whether you can that'd be great to know. Day one patches are at least as much of an issue on Xbox One though, for example I think Forza adds a notable amount of content with a patch including, I believe, the ability to finish career mode.