Do you spell out Thm., Prop., Eq., Ch. in mathematical papers?

It is a matter of style. I would say yes, expand them. In my opinion, authors tend to over-abbreviate making documents harder to read. For instance, I can't work out what you mean by Ch. (Conjechure?)

In my experience, authors almost always spell out words like "Theorem", "Proposition", and so on. I expect that journal styles will generally require this. I can't remember the last time I saw a published paper that abbreviated them.

But if you're writing a paper, you must have read a lot of other people's published papers. Surely by now you've formed your own opinion of the consensus?

To expand on Dave's answer slightly, abbreviations should follow the guidelines of the specific venue to which you are sending a paper. If they expect no abbreviations, don't use them. If they have standard ones specified, use those as appropriate. Typically, I would only use something like "Prop." for "proposition" when it's referring to something with a specific number, and that's what the style guide calls for.

Other abbreviations should be used to improve readability: for instance writing out "fast Fourier transform" one hundred times during a paper can start to get more tedious than using "FFT" as an abbreviation. But shortening individual words should only be done if it makes reading the paper easier, not simply to shorten it.