Small gauge wire mesh under your bin will keep rats out. Even the odd teabag cover can be riddled out on use.
Equal measures green and brown waste - egg boxes, card, good place to chuck shredded paper. Avoid shredding window envelopes.
Ants are good, so don't worry if they appear in your bin. Break down waste beautifully.
As well as tomatoes, pepper seeds seem to survive well.
I have a 3 bin system - lots of hedge and lawn clippings. With an additional bin for storage. Bin one gets full is turned into bin 2. Bin 1 full - bin 2 turned into bin 3, 1 into 2. Bin 1 gets full - bin 3 riddled and anything not broken down yet goes into 1, bin 2 to 3 and 1 to 2.
Got it down to a fine art now that the digging over just takes a couple of hours twice a year. Riddling takes a lot longer.
I don't worry about bits of egg shells gives slugs/snails something to nibble so they leave my plants alone!
Haven't needed to buy compost for years.