Differences between 正在, 正 and 在 with the meaning of an action in progress? When should I prefer to use only 正, only 在 or both together? Are there any rules? Does it sound really weird or incorrect to native speakers if someone use them improperly?
正 in itself has many meanings, one of which is "just", and thus can be used in conjunction with 在.

正在 is what you are doing at this very moment, or where you are right now, or in general what is currently present, the progressive aspect (the English -ing form) if you will.

我们正在吃饭 = We are eating

西方正在经历的艰难 = The economic hardship that the West is presently enduring

我正在北京 = I am currently in Beijing (I just got here)

我在北京 = I am in Beijing (more or less stationary)

You could not use 正 alone in a similar fashion.

The progressive 正在 could also be contrasted with the continuous aspect using 着. 我正在穿衣服 = I am getting dressed (current action), while 我穿着衣服 = I am wearing clothes (current state).

Then it gets more tricky when you use both simultaneously, as in 我正在爱着你.