"I always have problem to properly calibrate my mouse when playing FPS games (Combat Arms)."
I know this answer is very late; but I've played quite a bit of Combat Arms over the years and I believe it's critical to OP's (former) issues.
Combat Arms in 2011 was using an engine similar to what is now available as "Combat Arms: Classic" (this is not the same as the one available on Steam called "Combat Arms:Reloaded"; which doesn't have these issues). In the old engine, there are a couple things to note about mouse movement.
Combat Arms had forced mouse smoothing (I can't find the reference for this anymore, it was said by a developer on Nexon forums when they were working on CA:Reloaded). The best way to help avoid major effects from that was to put a cap on the FPS that you could maintain constantly.
"My mouse is Razer Copperhead."
Razer Copperhead, like many gaming mice, has a polling rate up to 1000 Hz. Combat Arms doesn't like that. The aim will zig zag and not follow straight paths if the polling rate is above 125 Hz. Most gaming mice have settings to limit the polling rate in their software.
TL:DR The Combat Arms: Classic (or Combat Arms in 2011) engine does not work well for calibrating mice in general. The game has forced mouse smoothing so using fixed FPS is suggested. It also requires a polling rate of 125 Hz or below to allow the mouse to move in straight lines.
Update: If (for some reason) you do still want to play Combat Arms Classic with a 1000Hz mouse, I have found the workaround for that as well.
In your Combat Arms Classic folder there are two .exe's for the game. Engine and Engine2. Engine2 is actually a "beta" build that has been included with classic for awhile.
The Engine2.exe actually works to run the game, (and removes forced mouse smoothing) as long as the game is up to date (which means you need to run the normal launcher to update the game).
You'll want to make a shortcut for Engine2.exe. Then right-click on the shortcut and go to properties. Where it says Target, add -windowtitle to the end of the target. It should look something like:
"C:\Program Files (x86)\CombatArms Classic\Engine2.exe" -windowtitle
This shortcut will launch the game with the build which removes the forced mouse smoothing.
When talking about advantages and disadvantages, it is necessary to first set a context. Each champion have advantages and disadvantages in different situations. These are several situations in which an AD carry will have to be good at to become effective as an ranged AD carry.
Early Game Farming/Harrassing
Caitlyn is notorious for her early game harass and farm. Caitlyn has the longest range (with the exception of Tristana once she reaches lvl 12) in the game at 650 and is excellent at poking at her enemy. If she utilizes her passive Headshot correctly, she can put unexpected bursts of damage on her enemy. In addition to that, her Piltover Peacemaker is a long range skillshot that does not require targeting. This results in a very one sided harass from Caitlyn.
Miss Fortune on the other hand has a much shorter range at 550. However her skill Double Up is also very good at harass if you are able to position correctly. This makes her harass much less reliable though. Her skill Make It Rain isn't a very good harass because it deals damage over time only if the enemy is within the area where this skill is casted. The enemy will rarely stand still in this area so it will result in a very negligible amount of damage. Not to mention the fact Make It Rain is a skill that is usually maxed last. However Make It Rain is very good at last hitting multiple minions at once if you find there are several minions that are about to die and you can't possibly simultaneous get all the last hits.
It should be noted that Miss Fortune's passive Struts gives her up to 70 additional movement speed after 7 seconds of not getting hit. This can be utilized to position herself to use Double Up so that bullet bounces to the enemy Champion. The speed boost is also very helpful to dodge skillshots such Caitlyn's Piltover Peacemaker.
Escaping Ganks
Inevitably during a game you will be ganked at one point. Because a ranged AD carry is usually very squishy, having the ability to escape and avoid damage is almost a must on an effective ranged AD carry.
Caitlyn is very effective in this respect. Her Yordle Snap Trap if placed in strategic bushes can warn her of an incoming gank and give her time to fall back to her turret because it provides vision and immobilizes the enemy that sprung the trap for 1.5 seconds. In addition to that, the initial cast of the Yordle Snap Track will very briefly provide vision inside of a bush. This can be very useful to scout whether there are additional enemies in the bush or not.
If somehow the ganker was very sneaky and comes undetected, Caitlyn's 90 Caliber Net is a great escape tool. It shoots a net at the enemy to slow them AND shoots Caitlyn backwards. This in addition to flash(if she has that as a summoner skill) will widen the gap between the ganker and Caitlyn.
Miss Fortune unfortunately lack such utility. Her only skill that can aid in her escape is her Make It Rain which damages and slows enemy champions in a small area. However, note that her Make It Rain can also briefly provide vision during the duration of the skill if cast in a bush. Although to be fair her passive gives Miss Fortune up to 70 bonus movement speed. This is lost however if she is hit even once(this will mostly likely happen during a gank).
Damage Output during 1v1
Although at first it may seem like Caitlyn has all the advantages, it is not quite the case in terms of pure damage output. Because two of Caitlyn's skills are utility skills, she has less burst than Miss Fortune. Miss Fortune has a built in steroid skill called Impure Shots which passively added additional magic damage to her auto attacks for each attack that stacks up to 4 times. When activated she will attack 30-50% faster and cause her enemy to heal only 50%. This is terrific for 1v1 another AD carry who will most likely rely on their life steal to stay alive. In addition, Miss Fortune's Double Up skill will reset her attack timer. This means if she auto attacks once, uses Double Up, she will immediately attack again regardless of what her attack speed is.
Caitlyn on the other hand has an ultimate called Ace in the Hole which is VERY effective at securing a kill from an enemy who is running away with low health. Or for chipping down the enemy's health before they are even close enough to hit you. Caitlyn's Headshot is great for 1v1 especially if the fight takes place in the bushes. She can effectively deal 150% every 3-4 attacks.
Damage Output during team fight
Although Caitlyn's Ace in the Hole is great for single target, it isn't very effective during a team fight. Ace in the Hole can be intercepted by another Champion(usually the tank who will take much less damage). Miss Fortune's ultimate Bullet Time on the other hand is very effective and lends itself great to the play style of a ranged AD carry. She is able to stay way in the back and as soon as the team fight starts, she can use Bullet Time's AOE(area of effect) damage to hit all the enemy. Then after most of them are low, you can come in and take the kills.
Caitlyn's Piltover Peacemaker does do AoE damage in a line, however each consecutive enemy hit by it gets reduced damage so that it really doesn't end up being quite effectively. She is more likely to do more damage with her autoattacks at that point.
Other Factors
Other factors to take in account is the statistics of the Champion without any items. Most of their base statistics are very similar but the main difference is their base health. Miss Fortune will start out with a base of 520 (435 + 85/lvl) health while Caitlyn will only start with 470 (390 + 80/lvl) health. In addition, Miss Fortune will continue getting 5 more health per level than Caitlyn will. This makes Miss Fortune slightly tankier than Caitlyn with respect to base health.
Best Answer
Well it's pretty simple:
You are faster (Watch the mouse movements of Wildturtle for example)
The cool thing is: The faster you move your mouse, the better your overview of the game is. You can basically "scout" The whole map with a fast mouse and then just hit space again to focus back on your character in under a second (you have to be able to realise what's going on in such a short amount of time though)
Of course it's not just like "Hey i'm gonna increase my mouse speed and then I'll reach diamond". That's where the second point you're asking is important:
The biggest disadvantage is you will play awful if you aren't used to it. You will missclick all the time and you will move your cursor too far if you have a sensitivity that is too high. It's also something that you can't learn in a short amount of time. If you want to go up for 1000 DPI it can really take a while (weeks maybe even months depending on how fast you get used to it). Also for practicing with increased mouse speed I wouldn't recommend MOBAs but Shooters or Games that require a high accuracy and speed (Osu is a great game for increasing mouse speed but usually shooters are more fun to play). Of course you can also practice it in League or other MOBAs.
I recommend going up only for about a maximum of 10% of your current speed at a time. Mistakes won't happen that often (they might still be game changing). And don't get frustrated if you make mistakes. It's completely normal and as already mentioned it's really hard to improve your mouse speed.