I'm new to sniping and myibay, and despite having read the FAQs, I need to be clear about: To use the free service, I need to be online at the time of auction, yes? And if I don't have a stable or fast connection, I shouldn't depend on my snipe succeeding?
In which case, I should download the shareware software, right? It seems there are two different downloads (not including updates), one the free version and the other, shareware (stand-alone, server supported, etc.)? Or am I missing something?

You don't need to be online:
http://www.myibay.com/login/faq/2/#q9
I guess that also answer to your question that your internet connection has no relation to your snipes.

The paid version is only for the Windows application which helps you to manage your snipes. This is good especially when you have dozens of items in your list.
If you snipe occasionally (once per day or less), the free Web-based interface should be enough for you.

Any version (free and paid) does not require you to be online during a snipe.

If you feel like you want to support the sniper, you can buy the software or donate other means (you can check Donate page for more info), but this is not required.