If you do all quests (side quests as well as the main story quest) you will get all the gear you need while leveling. At level cap you will need to worry about gear, but that is a long way off.
One which suits your timezone, geographical location, and language.2. What's a good realm to be on? I realized today I'm actually on a Japanese server.
There are three groups of data centers, one in each of Japan, North America, and Europe. It is typically recommended you pick a server in a data center close to where you live for best performance.
Yes, they are pretty rigid. But you can pick up any and all classes and jobs on single character and thus switch between roles that way.3. Are jobs pretty rigid? i.e. white mage is always a healer, and paladin is always a tank?
Mods are actually not allowed at all for FFXIV. There do exist some, and using them will probably not get you into trouble, but all of them are really disallowed according to Square-Enix's rules.
4. Are mods really a thing for ffxiv? Tried a quick google search but they mostly seem for aesthetics.
FFXIV is not FF XI. Races make very little difference in game play in FFXIV.5. I tried ff xi a long long time ago, and tried to play a taru taru warrior...and it was so hard to get an invite from anyone. Is that still the case or are all race/job combinations acceptable?
Definitely! All quests are meant to be done - but not all need to be done at once.6. If I joined the gladiators guild, but then when I went to other guilds there are quests available, is it ok to take those quests too?
Like I said above, you can pick up all classes and jobs on a single character.
At start - at least until the main story leads you away from your starting city - it is probably best to concentrate on a single class, but should you start to outlevel quests you should feel free to switch between classes/jobs to share the XP between them.