'To take a shower' is the standard American way to say it. If you want it to sound more informal, you can use the contraction 'gonna' instead of 'going to.' It's very common to not enunciate 'going to' and instead it turns into something sounding like 'gah-nuh.' 'I'm going to (gonna) take a shower.' You can also say: 'I'm going to be (gonna be) in the shower,' it's like the prepositional case way of expressing this. Both ways are fine; maybe the second way is slightly better in this context because it emphasizes that you will be unavailable and in another place (i.e. in the shower and not at your desk) whereas the first is more emphasizing the activity. As another example, if you couldn't do something in the context of being busy due to a vacation, you could say: 'Sorry, I can't, I'm gonna be on vacation.' In US English, we'd say it this way rather than 'I'll be vacationing.'

It's technically valid, but could be more natural. It is natural enough for a formal tone, but it seems like it should not be using a formal tone because it's revealing intimate personal details. Non-native speakers have extra licence to speak formally where native speakers would typically not, so it's not a big deal. Bit if I were to express the same thing, I would say it something like this: I'm only asking because I'll be unavailable for the next half hour. I'll be having a shower soon. This now has a warmer, friendlier tone, and "rolls off the tongue" easier. It's not so cold and formal like your example was.

Yes, it is correct and natural enough. I would say it this way: "I ASK this because I will be unavailable for 30 minutes. I AM TAKING a shower." Note the use of the simple present tense, "I ask". I try to avoid present participles when they serve no real purpose. "I am going to..." explicitly places the action in the future. That is certainly correct since the shower is a future activity. However, it is not necessary to do this. "I am taking" describes future action just as well as present action. Just as you can say "I am swimming" when you are nowhere near a pool, you can say you are "taking a shower" even though you are dry. The fact that you are talking to someone is sufficient proof that you are not yet in the shower.