The name "factory reset" is a bit misleading in my opinion. "Factory data reset" would be more accurate.
All a factory reset does is erase user apps and their data, and reset any system settings to their defaults. It does not alter the ROM in any way, which also means that any modifications you have made to the ROM, including rooting it, deleting system apps or adding apps to the system (as opposed to just installing them normally as user apps) will still be there after the reset.
Now as Scary says, there is a slight ambiguity in your choice of words. So if you meant restoring to factory software, this is done using a ROM Update Utility (usually referred to as an RUU). This is a Windows executable which will reflash the phone with a complete set of official software - you basically connect the phone via usb and run the RUU on a PC (if you are running a non-Sense ROM you'll have to put the phone into fastboot mode first - I'd probably do that anyway).
I believe that you have to relock the bootloader to run an RUU anyway (of course you can unlock it again afterwards if you want). Running an RUU will unroot the phone and give you a stock ROM (it replaces your current one with the stock one, and since root is a modification to the ROM and the stock ROM is unrooted, this means you lose root). It will also replace your custom recovery with the stock one, and will do a factory reset in the process. Which software version you will have will depend on what RUU you ran, but you would then be able to download any official updates that were available.
So in short, if you mean factory reset, the answers to all 4 questions are "no". If you mean restore to stock with an RUU, the answers to the first 3 are "yes", and the fourth is "if there is an update available you'll be able to install it".
Couple more things to note:
* RUUs check the "cid" (customer ID), so you need to find the right RUU for your phone. This will depend on region (which part of the world the phone was sold in) and network (if the phone was sold through a carrier). Also the RUU is an upgrade
utility, so will not let you downgrade your software to an earlier version, so if e.g. you updated to 4.2.2 and then rooted, you won't be able to use a 4.1.2 RUU to unroot.
* If you wish to stay rooted but want the latest updates, the best way is probably to just use a custom ROM based on those updates, rather than unroot, install the update and root again. These are usually available very shortly after the updates are released - some developers will use leaked updates and so have their ROMs updated before the update is available officially (e.g. the 4.3 update is only just beginning to be rolled-out, but ROMs based on it have been available for weeks).
Hope this clarifies things.