Heyho!
1. We usually say that the best way to play G&W is a passive-aggressive G&W, in which you stick around the opponent to lure out stuff. There are some MUs in which you have to convert to a whole different neutral, but that's usually the way to go.
2. Simple: You don't! Well, in the neutral at least unless we're talking about Dedede. Almost the entire cast can punish chef easily in the neutral. If you're in the advantage state (as in opponent is off-stage, whether above or jut away), you can use it to put some pieces in his way. But that's about it.
3. I take it that this is one of ROB's worse MUs. Good for you, because the Mario MU is even-ish, if not slightly in our advantage. Mario can't heavily combo us (although with platform assist yes he can), he shouldn't edge guard us with cape/fludd/fair/whatever, and so he has to poke around and then read for the kill (if he aims with his bair low enough he can beat upsmash). Sounds annoying, if G&W wouldn't have disjoints, annoying range as well and over-all a better defensive moveset. Upsmash and back air will deal with his aerial approaches, the presence of bucket alone will make the think twice to shoot fireballs (although they still can use two to get free punishes) and we can escape his downthrow stuff with our upB as much as he can (Mario can use upB to break our dthrow combo's at low%, be aware of that).
In a nutshell, both will bait out stuff, but G&W will usually be able to re-poke him for poking and get more out of openings unless you give Mario platform assist.
4./5. His only kill confirm out of a throw is the Toot Toot, his dthrow upair. The bad part about it though... it's percent window is character depending and can reach from 20% to something as low as 5%. Also, the move is pretty inconsistent at the higher border percents. Yay. ._.
You still want to learn it though because like I said, only kill confirm out of a throw.
If you get past that percent range, this is where stuff gets annoying. You will have to read a whole lot then, preferably with a storng attack. He does have a few unreliable kill setups like upsmash to dtilt, but all of them are very risky in terms of lag (in this case dtilt being the lag thing. a fast character will be able to punish this hard, for example fox with his upsmash). Other kill confirms being landing upair to upair, or nair to downsmash require you to believe in them really hard. You basically ask for a punish if you hit his shield.
6. If you really want to learn this character, I invite you to the G&W Skype and/or Discord group. The G&W boards is pretty much a wasteland by now, and all of the major discussion happens in one of those groups.
Hope I could help! :3