First of all, what sort of thing is a punishable offense?
Ignoring my health is my husbands pet peeve. He is right, I just don't care, so he has to do it. Sitting in doctors waiting room with a severely spanked bottom is not fun.

Every now and then my teasing turns into pestering. Punishments for that can vary on how creative he is feeling. Usually they involve mean, crappy boring things like positions, wearing a gag, bondage...things that are fun for him, lol.

Sometimes I just get moody, irritable, bitchy without being even aware of it. He knows what I need though, how to handle me. I need him to be rough, strict, merciless, edgy - painful.

There have to be some things that are let to slide, no?
Normal couple negotiating - what school we are sending our kid in, what country do we live in. He cant use force to get his way in situations like that.

Other than that, even if I think I don't deserve it, but he explains his point of view and we talk it out, I agree because if he needs it I need it too.

I also wonder about the nature of punishments. Is it still "play" or are punishments serious facilitators through which to express anger?
There is play. There is half-play, he doesn't want to stop playing so makes it more edgy, its still a game, but serious enough for him to get his point across.

When BDSM punishment becomes an excuse for a Dom to express his anger it becomes abuse. If he is truly angry he waits until he has calmed down to deal with an issue.

Do people in D/s relationships have just regular-old-relationship fights (yelling and heated conversation)?
I can speak only for my husband and myself - yes. We banter a lot (its sort of a foreplay for us), have heated debates, but neither of us is a yeller or prone to cursing. We are always respectful and mindful not to say, in the heat of the moment, something we don't mean but that is impossible to take back later. It is easier to forgive and forget a mean beating than it is a hateful word.

What factors does one use to decide what the punishment will be?
"Rules" that are important for Him that I follow, (within reason). Its important to discuss at least general parameters of those rules when beginning a relationship, so she knows what to expect.

But to be fer he/Dom can never be as hard on myself as I am. Sometimes I ask Him to punish me because I need it (for whatever reason).

Do Doms breach "hard limits" of their subs when punishing them?
Not good Doms. Hard limits can be discussed. Some will always be off limit, others can be pushed and stretched, but punishment isn't the time to do it (without prior discussion).