Hey, congratulations on your purchase. You'll love it if you just keep in mind that Leopard can be quite different from Windows, but there are some similarities too.
1). Most Mac apps don't go full screen when you hit the green button on the left in the window. It's not a maximize button, it will only open the window just enough for the content within it to be visible. There are a few different reasons and theories for this, but the bottom line is most apps will require you to drag the window corner to get it to go full screen. But the app will usually remember that window size and keep it that way the next time you open it, and hitting the green button toggles between the user-defined window size and the standard size.
2.) Choose History>Home from the menu or use the keyboard shortcut, Shift-Command-H. Shortcuts are used extensively in Mac apps. I don't know why there isn't a Home button in Safari, but then I use Firefox. I would check the Help menu of any app if you have questions. It usually has some good answers, that's how I found out how to get to your home page just now.
3) Yes. If you have a mouse you can right click for contextual menus, even an Apple Mighty Mouse, which looks like it has just one button but it actually has four. If you are using your trackpad, you can enable it so that a double-fingered tap acts as a right click. Just go to System Preferences (under the Apple Menu which is always at the far left of the menu bar) and choose Trackpad. Where it lists the Two Fingers gestures, check the "Secondary Tap" box. Another way to "right click" and bring up contextual menus is to click on something while holding down the Control key, known as Control-clicking.
4) Is the wireless connection password-protected? If you click the Airport icon on the right side of the menu bar and see a little padlock icon next to the connection's name, you'll need a password.