Is tungsten a good metal for jet engines, besides the weight penalty? Right now, the hottest part of jet engines are made from a steel-nickel alloy. Is tungsten a better choice?

Pure tungsten has the highest melting point and highest tensile strength of any other pure metal. Presumably you could make it hollow and use the same cooling fluid to prevent overheating. For this same reason I am not sure if the weight penalty is really that big. I don't think the turbine blades make up a really big part of total engine weight.

(I don't know how well it resists oxidizing at high temperatures. I don't know how it might change with adding a little carbon or steel or nickel. But presumably this is not a drawback because I've read about rocket engine nozzles using tungsten: link. If it doesn't oxidize or corrode there, I can't imagine why it would do so in a jet engine.)

Another way of asking this question: Does tungsten in the hottest part of jet engines have any major drawbacks besides weight and cost?
No, tungsten would generally not be a good choice for jet engines.

Some of the parts that would need to be the most heat resistant, such as turbine blades are subjected to rotational forces that increase directly with the mass of the part. Thus a high mass material such as tungsten would increase the forces the part would have to withstand. These forces are already measured in tons per blade.

If you discount the high-mass quality of Tungsten, the biggest problem is that Tungsten is also brittle. If a turbine blade failed, as does happen, it would be very difficult to contain. Tungsten is used in armor piercing projectiles; the results of a tungsten turbine blade failure would not be good.