[wayrad]

My device uses an older version of PalmOS than yours, and requires an aftermarket card and driver for Wifi, so I can't answer all your questions, but I'll try to help! Hopefully the things I am describing are on your OS version too...

Connections is where you tell the Palm what hardware you want it to communicate with, what technology to use (eg BT, wifi, cradle/cable, IR), what data speed to use, etc. You need a Connection for each device you might want to tether to. Some are already in the list, like Cradle/Cable; some you need to create.

Network is where you enter settings that relate to getting in to your ISP or phone provider's system. You may have more than one Network configuration also, if you plan to do different things. One of the things you select when setting up a Network is which of your Connections you want to use to make the actual connection. Network settings are not relevant for hotsync AFAIK (bear in mind that I have no experience with wifi hotsync).
There is a "Connect" button on the Network screen which is useful for testing whether you can actually reach your ISP or phone provider; however, you will be able to do nothing with the resulting connection since it isn't going through your browser or email app. Those apps will give you a place to select one of your Network profiles (which in turn specifies one of your Connection profiles) and normally you make the Internet connection through them.

Phone is a screen where you can select one of your Connections for use in "phone applications". I have only found two things this is useful for - one is to select a BT connection to the phone and use it make the Palm dial numbers in your address book for you (via the "dial" command in Address Book). I also have a little app that lets me check signal and battery strength on my phone from my PalmOS device, and it apparently uses this route. The Phone screen seems to have nothing to do with hotsync or internet access.

Bluetooth is where you pair your Palm with other devices. If you have trouble seeing a device, make sure it is in "discoverable" or "visible" mode. The reason you paired with a phone and still don't see it in your Connection list is probably because you haven't created a Connection for it yet. Once you create a Connection it is used to populate the appropriate lists from which connections are selected in Network, Hotsync, Phone, etc.

Infrared is useful for "beaming" information between two Palms, for syncing to an IR-enabled desktop, for printing to an IR-enabled printer, getting online with an IR modem, running a TV remote app, or entering information with a folding IR keyboard. Some of these uses have been rendered more or less obsolete by Bluetooth. You should definitely turn off "beam receive" when not in use because it is a major battery hog. (The other main battery hog is the screen - turning brightness down will help battery life a lot.)

Hope this helps.