1. Why do Runescape have this rule?
Runescape Gold have this rule to maintain a fair gaming experience. We dont want our games to become games where everyone feels they need to use multiple accounts at once just to compete! It‘s much better to team up with your friends.

 2. Can I use third-party software to log into an account for me while I am logging into another (so that I personally am only using one account)?

No, you may not use third-party software to gain an unfair advantage - see macroing and use of bots or third-party software.

 3. I have two accounts. Can my friend play one so I can trade between the characters?

No, this goes against this rule. You are not allowed to trade between two characters owned by one player. You would also be in breach of the Account Sharing/Trading rule.

 4. Can I give items to another player from one of my accounts, so that they can then give the items to another character that I play?

No, this goes against the rule. You may not transfer items between your accounts in any way.