I couldn't find any porn by searching for all the terms together - cat, home, soup. It's a bit sad when people feel the need to lie to convince people to join their cause. This one seemed comically implausible because the outcry would be fantastic against this kind of thing (can you imagine - the Super Bowl a few years ago showing a nipple - and catching your kid finding porn when they just want to see a cat?). Most search engines have taken measures exactly to prevent this from happening, and it seems to be very effective.
Yes. Search engines, even fringe ones like the one I use, have "safety modes" enabled by default that you have to turn off yourself. I'm pretty sure you can also attach passwords to them.