My first question is about whether or not people that aren't users of the site are able to wish in users' wishing wells? And if they can wish in users' wells, would they be allowed to do that?

My second question is about what all the achievements are and how I could get them?

I'm not sure about your wishing well question, but the other I CAN answer.
The first set of achievements are for how much money you have in the bank.
1. 100 million
2. 500 million
3. 1,000 million (1 billion)

The next set is for how many coins you have carried at a time.
1. 1,000
2. 5,000
3. 10,000

The next set is for your trainer level.
1. 50
2. 100
3. 250

The next set is for how many bets you've won. (From playing the 50/50 Game)
1. 100 bets
2. 250 bets
3. 500 bets

The final set is for how many times you've won the Daily Raffle.
1. Once
2. Twice
3. Thrice

Yes it is possible for non-members of Creed to wish in others' wells, as long as they have the specific link for the well, which in this case is: [you must login to view links], where xyz stands for the trainer's Trainer Number. However there is a limit to how many times a player's well can be wished upon per day, including once per IP address, account etc.