Rooting gives you access to areas of the device specifically "/system/" level areas that the OEM/carriers lock down. Yes it does void your warranty but only while in the rooted state. That is why there is normally a restore to stock method in the event you no longer want to be rooted or need to have the device serviced/sold.
To answer your questions..
1. A ROM is a unofficial version of the official OS. These are created by popular ROM developers like Cyanogen Mod team or AOKP (Android Open Kang Project) and others based off the official OS code and modified to their own standards. They are fun, normally very stable and offer many cool mods in the form of user selectable/customizable options over stock.
2. Flash counter is put in by the OEM to keep a count of how many times you have " flashed " the device. In other words, specifically Samsung devices whenever you use the app " Odin " which is used to flash the device to gain root access or flash custom software it counts as a " flash " on the flash counter. There are work arounds and hacks to reset this.
3. Before rooting, make sure you have backed up any pictures, music, etc. from your device in the event it is erased. Also make sure all your contacts, calendar, etc are sync'd properly with Google & your Gmail account so it can be retrieved after the rooting process has been completed.
4. After you root it is up to you what you want to do. Some use root on the stock OS so they can tether for free. Others use it to remove the OEM/carrier bloatware. Then you have custom ROMs which can be flashed too.
Best thing to do is read and research the methods & terminology associated with rooting and ROMs so you are more comfortable and informed before attempting the process. You do that and you will have a much safer and successful experience.