Welcome to the hobby, as others have said, you can use the shift functions to eliminate specific buttons and still have the inputs available. I will try to answer your specific questions below:
1. Can I get by without an enter button? When I use a sidewinder joystick, button A seems to function as an enter button so I'm wondering if I really need it.
Depends - I use EmuLoader for a frontend and the only way to launch a game is enter or a mouse click. In MAME, if you disable startup screens, then you can move the joystick left and right, or run NoNAMEMAME and disable all nag screens, and then you don't really need enter at all.
2. Could I wire Coin1 and Coin2 together to get an extra button to function properly or are there certain games that this might cause problems with?
Will cause problems with games like Gauntlet and Quartet which were looking for individual coin slots.
3. If I can get away without using an enter button, I'm thinking I'll probably just wire up the Tab, Esc, Reset and Pause buttons, does that make sense to you?
Tab - Personally, I would prefer to leave this off - Only required for config changes and I prefer to swap the I-PAC for a keyboard and do that and then swap the I-PAC back in so only I and not accidentally someone else can mess with my MAME setup.
Esc - Yes, I recommend this one. If you want to save space or inputs, set Pause as your shift key, and then Pause + P1 Start as Escape.
Reset - I assume you mean F3 - Can be useful if you don't hit it accidentally, but it's about as easy to escape to the frontend and restart the game. I wouldn't go out of my way for this one.
Pause - If I could only have one admin button on the panel it would be this one, I would disable the Coin 2 before I left it off, but you could activate it via a shift function.
4. Is the service button really necessary for these games? I've only used it once for Joust and that's about it.
Service for the Williams (Joust) games is F2 - you will press it the first time you launch the game and then you can forget about it. Like Tab, connect a keyboard, set up the game, plug the I-PAC back in.