Don't worry, you are not the only one who has had problems getting all the patches and things sorted out.
I found this
written installation guide very useful. It covers Steam and retail installations as well as other stuff like using patch updaters etc. Alternatively, this
video guide also seems to be quite good, although it gets a little more complicated toward the end when he starts talking about an ArmA2 launcher. At this stage, you don't need a launcher but you may find it useful once you've got your head around the whole deal.
I'm not sure if you have the Steam or retail version but either way you will need to make sure that both ArmA2 and ArmA2:Operation Arrowhead (which are collectively referred to as Combined Operations) are installed in the same
directory eg. C:\Program Files (x86)\Bohemia Interactive. I run the retail version so if that's what you have, you may find this useful. The principles are essentially the same for the Steam version I think but I don't know the specific in's and out's, sorry.
You will need to run ArmA2 once it has been installed (just to the menu screen then quit out) and also ArmA2:OA once installed. If the game has recognised that you own both ArmA2 and ArmA2:OA, it should show ArmA2 Combined Operations in your Windows Start menu. When you run ArmA2:OA, you will also notice that there is an ArmA2 and an ArmA2:OA icon in the bottom left corner of the menu screen. If this is the case, then you are heading in the right direction. If you do not have either of these indications, chances are that your game has not recognised that it is Combined Ops. If so, you will need to go back to your directory and make sure that both games are installed in the same
place. This alone seems to be one of the biggest issues in people being unable to get this game up and running.
Once you have installed and determined that Combined Ops is recognised, go to this link
to download the most current patch. Your game may already be patched to the latest version, but then again it also may not be. I bought a digital copy from Amazon and it still required the patch. The patch is pretty straight forward, it's a matter of download it and then run the .exe file. The patch will locate the directory automatically and install itself.
The installation guide should be able to help you sort out the other stuff. In regards to your graphics, did you purchase ArmA2 and OA or are you using ArmA2 Free with OA? If you are using the F2P version it will result in the use of low res textures and such, which may explain why your graphics aren't great. If you purchased ArmA2 and OA and still have bad graphics then we may have a problem. You also mentioned your computer not liking the 'sprocket' thing, which I'm thinking may be a Windows Command Script. Did you try to install the beta patch for ArmA2 at some stage?
Either way, everything is fixable so try not to stress. Have a look at the guide and if you are still having troubles then post back here and I'll see what else I might be able to help you with.