Of course rats can be anywhere, but provided you only put green waste from the kitchen and garden onto your heap there is no reason why rats should be attracted.
I've never had rats in any of my 6 compost bins. However, rats love decking, and my neighbours had a family under their's! Apparently the rise in decking has been far more attractive to rats than compost heaps!
Not sure if its the same company but I send all my plastic periodical wrappings from work to a company called:
RR Flexco,6 - 10 Concorde Road,Norwich,Norfolk,
I've been composting for many years with no real problems with rats although I'm lucky with a high cat population which I assume keep things under control (just) - I do know the London borough I live in does have a rat problem.
With regards as to what goes in - well everything!; from the obvious garden waste (shredded if need be) to the dirt collected in the hoover. Even breakfast cereal boxes go in with any shredded confidential waste - it just has to be left long enough (9 months for me) and kept damp.
We were given a compost bin years ago by the local council and it quickly fills with the usual kitchen scraps. Then about four years ago they provided wheelie bins for garden waste to be collected fortnightly. The curious thing is that they insist that only green waste go into them - no lumps of wood and definately no cardboard/paper. They also provide boxes for newspapers and magazines, glass and tins - but not cardboard. So I am pleased to read that it can go into my own composter rather that burn it or bin it with the household waste.
Re E White finding the egg after two weeks - we found one after umpteen months when digging a border and it was not a pleasant experience!