Dachalover, you are right about dog owners who fail to train them or take necessary precautions to stop them causing problems. As for the idiots that train dogs for use as a weapon or use them for fighting, the government should make it legal to shoot them. But let's not stray from the topic (cat debates always end up as cats Vs dogs arguments).
Cats very rarely destroy lawns or trees.. just brutally murder wildlife such as birds (their speciality), rodents, not just mice or small rats but shrews and voles, frogs, toads, butterflies an no doubt many other creatures.
Cats certainly crap on lawns a lot and have a tendency to crush, spray urine on or dig up flowers in neighbouring gardens as a way of marking territory. They tend to do it only in their own gardens when competition for boundaries is fierce. That's probably why you experience few problems in your current garden.
Fair enough, every animal needs to excrete poo somewhere but when you've lived in different houses over the years and in each one, cats have destroyed beds of newly planted flowers, or every time an inch of soil is disturbed during weeding, borders are turned into an irresistible litter tray for weeks on end. Then when the borders are hard or too full of crap, the lawn is the next target, then when they got bored of lawn, back door steps seemed a delightful toilet facility.
I'm now a part-time gardener, not RHS qualified, just someone who does a thorough job. My biggest menace is cats crapping all over customers' lawns, in borders, on paths etc. Some young mothers are worried about letting their children play in the garden due to the mess cats leave on the lawns.
And the irony is that only two customers own dogs (and they clean up after them thoroughly) and only 1 customer owns a cat and there has never once been a pile of cat filth on his massive lawns.