Learn English – Expressing similarity in different aspects

I have no room for other things in my mind.
I have no space for other things.

Both of these are figurative speech, and they are acceptable as is. One might add the word extra or spare as well:

There's no spare room for anything else in my mind.
I have no extra space for new things in my brain right now.

(I decide to change the way you said other things, because it's hard to figure out what "things" are in your mind already, and what other things you don't have any room for.)

But, yes, you can use words like space when talking about the brain for the mind. In fact, the word attic is sometimes used to refer to the mind metaphorically, because it's in the upper part of our body, and it's used to store things. Doyle used this analogy in 1887, in a Sherlock Holmes story:

“I consider that a man's brain originally is like a little empty attic, and you have to stock it with such furniture as you choose.” (Sherlock Holmes, A Study in Scarlet)

As you guessed, words like space and room can also be used for a schedule, as well as a brain:

I'm sorry, I'm too busy; I don't have room to meet this week. Can we try for next week? I have more free space on my calendar then.

In addition to brains and calendars, you could also use such words to describe, say, a person's frustration level. For example:

My car won't start! Dammit! I don't have room for this right now.

might mean that the speaker is under a lot of stress, and this isn't a good time for his car to break down. Here's another example from a book:

As much as I don't have room for this in my life right now, I can't help but be intrigued. “What kind of plan?” I ask. (N.L. Shepherd, Stealing Bases)

When you put an apostrophe, it can only represent two things:

1. Possession, e.g, Alex's bag.
2. To shorten a word, e.g haven't.

If you were to say "utility's" it would mean that the utility is owning something. I assume that that's not the case, and you were talking about the pilot utilities, plural form of utility, without an apostrophe. Thus, utilities should be the correct word to use.

You said, "one of our", which means there is more than one utility. If there is only one utility, you would say "our pilot utility".

Finally, the correct sentence would be:

Have a look at the impacts in Cologne, where one of our pilot utilities is located.