1a,3a: Urgot. Poke with long range, if they come in close then just eat their face. Q + Auto Attacks = death. You have a shield, and you lower their damage. If they stealth, guess where they are with your Q. Keep in mind that you don't need vision to auto-lock a Q on someone who has been hit with E.
2a: You generally have to bring 2 people to fight Jax properly, as he excels at 1v1 situations. Urgot does okay against Jax, but his ult works really well for shutting Jax down long enough for someone else to kill him. Teemo can do okay against bad Jaxs (my friend insists Jax completely destroys Teemo every time, personal experience says otherwise) you just have to be smart about your shrooms, w, hextech, and exhaust. If you blind him, do it after he leaps, and take that time to do as much damage as possible or to gain as much distance as possible.
1b,3b (if AP): Odyns Veil, or a health item. A little bit of defense goes a long way vs both Shaco and Akali. Vision items, including oracles, are also a huge help.
2b: Sword of the Divine is okay on some champions, but is generally a bad item. DFG, Thornmail, Sunfire (kind of), Hextech, and Odyns all help. Specifically, any item that does damage without risking a dodge proc works well. Obviously these aren't the best items for all champions, but they all have their place.
3b (if AD): Thornmail, Oracles.
EDIT: And as you said, Rammus is countered by map awareness. If you want a specific champ to help vs a Rammus, pick Teemo. Keep their speed shrine, your speed shrine, and the bushes above storm buff shroomed at all times (their speed buff can be hard to keep up depending on how the game is going, you might want to consider putting it in a place where it won't get stepped on).
EDIT2: Items listed are suggested items that help a bit individually versus the champions listed. They are not builds.