Is it unwise to start a Ms/PhD program without a stipend - will it upset professors if I leave with a masters to apply to funded programs?
I think that you are more asking, whether it is common to start graduate school without funding. Generally at the MS level, it is not uncommon for MS students to be unfunded. All depends on the school. Most schools that I've encountered would much rather fund PhD students since you get more out of the students.

Now at the PhD level - I would not advise a PhD without funding. Trying to live 4-6 years without a salary is difficult, and most people I've seen try this failed. There are enough schools out there that you should be able to get funding somewhere (unless you really have bad scores/grades and shouldn't be in graduate school).