Question

How is Eevee's happiness increased in Pokemon Emerald?

Answer 1

>Walk 256 steps with the Pokemon in your active Party. (+1 for all happiness values)
Give the Pokemon Vitamins [HP Up, Carbos, Iron, Protein, Calcium, Zinc, PP Up] (+5 0-99, +3 100-199, +2 200-254)
Usage in Important Battle [Gym Leaders, Elites, Champion] (+3 0-99, +2 100-199, +1 200-254)
Level Up the Pokemon (+5 0-99, +3 100-199, +2 200-254 per level)

That is copied an pasted from [Psypokes][1].
Their guide is for D/P, but the same holds true for Emerald.
The stuff in brackets is how much the happiness value increases for each action.
[1]: http://www.psypokes.com/dpphgss/happiness.php

Answer 2

If a Pokémon is caught in a Luxury Ball or is holding a Soothe Bell, all friendship-raising events will increase the Pokémon's friendship by one point more; friendship-lowering events are not affected. These effects stack on each other, meaning a Pokémon in a Luxury Ball holding a Soothe Bell gets two additional points per event

Walking 256 steps
+1
Using vitamins
+5 (Friendship 0-99)
+3 (Firendship 100-199)
+2 (Friendship 200-Max)
Gaining a level
+5 (Friendship 0-99)
+3 (Friendship 100-199)
+2 (Friendship 200-Max)
Using EV Berries
+2
Fainting
-1
Using EnergyPowder
-5
Using Heal Powder
-5
Using Energy Root
-10
Using Revival Herb
-15

Answer 3

Walking 256 Steps
Self Explanatory, really.

Vitamin Use
Any vitamin will increase a Pokemon's happiness. The vitamins, if you didn't know are the HP Up, Protein, Iron, Calcium, Zinc, Carbos, PP Up, PP Max and Rare Candy. If your friendship level is below 99, it will raise by 5. If it's between 100-199, it will raise by 3. If higher than 200, it will raise by 2.

EV Berries
The friendship level will raise just like it would if you used vitamins.
The EV Berries are the Pomeg Berry, Kelpsy Berry, Qualot Berry, Hondew Berry, Grepa Berry and Tamato Berry.

You can also trade to FireRed and LeafGreen to get grooming from Daisy, which will raise by 3.

If you give your Pokemon any herbal medicine, it will decrease depending on what herb it is.

If you want to track your progress, you can head to Verdanturf and in one of the houses you will find a woman who will rate your friendship. Here's what she will say:

"It adores you. It can't possibly love you any more. I even feel happy seeing it." 255
"It seems to be very happy. It obviously likes you a whole lot." 200-254
"It likes you quite a lot. It seems to want to be babied a little." 150-199
"It's getting used to you. It seems to believe in you." 100-149
"It's not very used to you yet. It neither loves nor hates you." 50-99
"It's very wary. It has scary viciousness in its eyes. It doesn't like you much at all." 1-49
"This is a little hard for me to say... Your Pokémon simply detests you. Doesn't that make you uncomfortable?" 0

^^ All that friendship rated stuff from Bulbapedia.
Check frequently to get an exact rating of your progress. Of course, 256 step will detain you from getting an exact rating, remember.

Good Luck. Hope I helped. :)
Source: Above Link