Final Fantasy VII – Train of Corel

E Skill learns new skills by being subject to those skills in battle. So it's a matter of fighting the proper enemies, not enough battles. It's also impossible until the end of the game, due to one being in the last dungeon. There are 3 "problem" E Skills, as it were, that cause some frustration. 2 because they can be lost forever and one that's fairly luck based.

1. Chocobuckle, granted by level 16 Chocobos near the Chocobo Ranch. They'll only use the attack when low in health, and the only way to get them to it without them running away immediately is to use another Enemy Skill, Lv. 4 Suicide. Level 16 ones are the only ones that can be hit by the skill, anyway. However, it's never quite guaranteed that they'll use it, so it may take some time.
2. Trine, used by the bosses Materia Keeper and Godo, as well as Stilva, an enemy in a dungeon that can only be visitted once. So once you have beaten both bosses and gotten past that dungeon, it becomes impossible to learn the skill. So make sure that when you fight the bosses, you wait long enough for the skill to be used.
3. Pandora's Box, used by the Zombie Dragon enemy in the final dungeon. This one is interesting because in any save game, the skill can only be used once. So you need to make sure you have every Enemy Skill materia equipped at this point, because if you run into a Zombie Dragon and it uses Pandora's Box and you don't have it, you've just lost it forever.

Past those, the rest tend to be a simple matter of fighting the right enemies and waiting for the right attacks. There are some buff spells which will need you to use the Manipulate materia on the enemy, such as Dragon Force, Angel Whisper, and White Wind. If you see an enemy using one of these skills, then use Manipulate and force it on yourself if it's available. Past that, the rest is all a matter of getting your face beaten in.

General strategy is simple - always keep that Materia on at all times! Then you will not miss the one-times and you'll always catch it when the attack happens in normal scenarios, rather than having to hunt the skill down amidst all the rest of the late-game stuff you need to do.

No. You can cheat with things like the game genie, but she cannot be "legally" brought back.