It's 100% ok for plain bearings to get warm in use. In fact many old ultra high precision machines were run for 1 hour or so before being put "on line" as they were adjusted to run at that temp. Synthetic automobile oil is an old and contentious subject in horology. I remember talking to a tower clock guy years ago who had gone to Mobil 1, and then reserviced a number of clocks for free after finding that the oil turned abrasive after a certain length of time. I haven't been able to find literature on it recently but there was some 15+ years ago. Auto oil is changed on regular intervals in addition to being filtered. The demands, as measuretwice spelled out, are very different. In a plain bearing watch lathe the oil goes in and goes out- "total loss". So really anything relatively thin will work. In a larger machine it's an issue, as motor oils are designed to keep contaminants in suspension to be grabbed by the filter, and most machine tools don't filter their oil. Contaminants fall out in the sump- which won't happen with a detergent based oil that keeps things in suspension. But back to watch lathes- almost any oil is ok, sewing machine oil is great, iso 10 or 20 hydraulic oil, awesome, 3n1 fantastic, Hoppes gun oil, killer, just no WD40.