Is мне stressed (as in, is it pronounced mnyeh?)? Is this always the case? If not, what are some examples? Also, what about other short words such as по?
The pronoun мне will be always pronounced as mnyeh, no matter if its stressed or not. There's no way you'd say mni.
If you say it in a flowing speech (say, for example, "мне нравится этот фильм") you have basically several options of stressing to report fine meaning of what you're trying to say. Normally if you just report the fact that you liked the film you pronounce "мне нравится" joined together putting primary stress on "нравится", so you have something like (mnyeh-NRUH-vih-tsuh). If you stress the word "мне" putting secondary stress on the verb you would mean that you personally liked it as opposed to someone else who didn't. The same applies to the rest of the words in this example.
По is a preposition and it is also unstressed most of the times and you should pronounce it as any syllable "-по-" in its unstressed position (-puh-).
Now thinking about the syllable -мне- in unstressed position, I'd say that I would always pronounce it like -mnyeh- (consider the following examples: мнемóника, камнепáд or сомневáющийся) but maybe there's a counter-example I can't think of.
UPD 2016-03-02: Transcriptions edited according to hints of @YellowSky
UPD UPD 2016-03-02: As pointed out by @DenisKreshikhin standard pronunciation of a -e- following a soft consonant in an unstressed syllable followed by the stressed one would lean towards -и- rather than -e-