HAHAHA! Something I'm knowledgeable on!
1) Natures each have their own effects, but any pokemon can have any nature. More to follow.
2) The little face that pokemon get after having the Pokerus indicates they have the virus, and are no longer contageous. Again, more to follow.
3) You need the National Dex for that, and I'm fairly certain you need to beat the Elite 4 before you can acquire that.
Any way, to elaborate on 1 and 2...
Natures affect a pokemon's base stats; they can raise one stat and lower one by 10%, or they can not affect their stats at all. The neutral natures are Serious, Hardy, and Docile.
The Pokerus affects stats known as "
EVs", or "Effort Values". This is a hidden value you need to keep track of yourself. The properties of effort values are as follows:
-Any given pokemon can have 510 total EV points.
-Any given stats can have a total of 255 EV points.
-For every 4 EVs a stat has, it will gain an additional point.
Thusly, to avoid having wasted EVs, there should never be more than 252 EVs in any given stat. For that matter, any amount of EVs should be a multiple of 4.
-Only 10 Vitamins can be used per stat; each vitamin gives 10 EVs.
-EVs gained from battling stack onto Vitamin EVs; Vitamins do not stack.
ie: If you had a pokemon that had gained 50 EVs from battling, it would only be able to consume 5 more vitamins. Basically, once a stat has 100 EVs, it can no longer take vitamins for that stat.
-Certain items can accelerate EVs gain. It's pretty straightforward how they add up.
*Macho Brace- x2
*PKRS- x2
*Power items- +4
*Vitamins- +10 (static- unaffected by hold items and PKRS)
So, a pokemon with the PKRS and a Macho Brace that defeats a Haunter (yields 2 Sp. Atk EVs) would equate to this:
(2 Sp. Atk)*2*2 = 8 Sp. Atk EVs
If that same pokemon instead had a Power Lens, it would equate like this:
((2 Sp. Atk)+4)*2 = 12 Sp. Atk EVs
Conversely, if it killed a Bidoof (yields one HP EV):
((1 HP)+(4 Sp. Atk))*2 = 2 HP and 8 Sp. Atk
There are guides around the net to show which pokemon yield which EVs. Personally, I use:
Bidoof- 1 HP
Starly- 1 Speed
Gastly/Haunter/Gengar- Sp Atk (1, 2, and 3 points, respectively. Budew as well, if you want a weaker level, but not as common. 1 point each)
Shinx- 1 Attack
Kricketot- 1 Defense (I think)
Tentacool- 1 Sp. Defense
The power items tell you what they give in their description.
IVs, or "Individual Values", work a little differently, but are also very important; the pokemon's descriptive phrase can give a hint as to what its best IV may be.
IVs are static; they cannot be altered. Once the pokemon is obtained, be it a recived egg, caught in the wild, or picked up from a deliveryman, its IVs are assigned. They can be from 0 to 31. Each IV is basically a bonus stat point. Unlike EVs being a ratio of 4:1 per stat point, IVs are simply 1:1.
Use sites like
http://www.serebii.net/ as encyclopedias for base stat info, IV calculators, and the like. Let me know if anything else needs explaining, I kind of dumped a lot.
gonk Edit: Almost forgot. Properties of the PKRS:
-Doubles all EV gain for the affected pokemon.
-Very rare to find yourself; usually it's easiest to have a friend infect one of your pokemon.
-Pokemon become infected by riding in a party with someone who is contagious.
-The PKRS is permanent and always active once the pokemon has it, but when the pokemon is in the party past midnight (whether the game is on, off, or in sleep), it is no longer contagious. To remedy this, simply place the contagious pokemon in a box. When a pokemon is no longer contagious, the purple PKRS status goes away, and a yellow smiley takes its place. 
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