That info you just posted does look like you may have a hard drive problem ("There was error with disk hardware"). If your HD is bad you should replace that before updating/installing anything else.
As I said I couldn't be sure it is an SP1 issue but I couldn't rule it out.
And BSODs are often caused by out-dated drivers so yes you should update your drivers.
... but if you are going to replace your HD there is no point in doing any of that first; in fact if your HD is failing you should back up all your data to an external device ASAP in case the hard drive fails completely and you lose everything.
Once you have the new HD installed and have restored your system then, in regards to your Windows Update question. Personally I don't like to have my computer run Windows Update any time it likes. I set mine to just warn me that it needs done. I think that is especially true of a laptop because it probably isn't on all night to update when the usage is low so it often tries as soon as you turn it on and it uses a lot of bandwidth on my wi-fi at a time when I want to be pulling email, playing games, surfing, etc and I want the bandwidth for my stuff not background updates. But yes you should periodically run Windows Update to get at least the Critical Updates (ie security patches) and if you feel you can't remember to do that at least once a month then you would probably be better off setting it to automatic so that you stay protected.