Correct. It could be water seepage into the horn relay or some fault or short at the horn button on the steering wheel itself. Any electrician worth his salt could check that for you.
Oil-wise, stick to the thinnest oil that won't burn off. Synthetic or not, depends on your pocket, but bear in mind some synthetics have much longer change intervals, so sometimes for $20 more per quart, you can get an oil that lasts twice as long. Economics.
See here for more info:
viewtopic.php?f=2&t=282499It would've been wise to find out from the previous owner what fluids they were using and when was the last change.