1. Do I have to run guiminer 24/7 to have more of a chance of solving a block or could I still solve a block if I run it for only 12 hours a day?

Every hash you do has the same chance of solving a block. Half as many hashes means half as many chances.

2. I'm a member of slush's pool. Does this mean I will never have the opportunity to solve a block and only get the fractional 0.10 btc every day?

Every hash has the same chance of solving a block. But just as you get paid even if you don't solve a block, you don't get paid extra for solving a block. By being a member of a pool, you divide the revenue for any blocks found evenly. (Different pools have different rules. Some might pay a bonus to a block solver. Check your pool's distribution rules if you're not sure what they are.)

3. Will I eventually solve a block or is it totally random, and therefore, may never solve a block? I have been mining for 3 weeks and have mined 2.2 btc.

It's random. Every hash has the same chance to solve a block. You may never do it. You might do it twice in the next minute.

4. Does the number of shares have an affect on solving a block? At the moment guiminer shows I have accepted 16422 shares. What does this mean?

When you "almost" solve a block, you provide proof that you have done so to the pool. The pool validates this proof and credits you with one share. The pool then pays out based on the number of shares you have submitted. Typically, one in every 4.29 billion hashes will generate a share.

5. The bitcoin client only ever shows a maximum of 62 connections. I have forwarded the ports in my router. Is that normal? Does the number of connections have an affect on solving a block?

When you mine as part of a pool, you don't need to run the client. The pool assigns you work, verifies proof of work, and submits a solved block should you generate one. If you solo mine, you do need to run the client, but the number of connections isn't terribly important. It's mostly just to make sure that an attacker can't hide the real network from you and that you have a current view of what's going on. 62 is more than enough for any application.