Learn English – Should I say on Fridays or on Friday

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)

To me, "of them" comes across as slightly wrong, because it's partitive. Consider:

• "There were six apples on the table; Susie ate two of them."
• "Susie eats apples often. This morning, she ate two."

A quantity of something implies part of a whole: Susie ate a part of the whole set of six apples. But in the second sentence, which isn't partitive, one would not ordinarily say "she ate two of them".

Another example:

• "My two sisters go to church every Sunday."
• "Two of my sisters go to church every Sunday."

The of construction in the second sentence clearly suggests that I have more sisters than two, whereas its absence in the first clearly indicates that I have only two sisters.

Similarly with your favorite colors, or the cars of your dreams: you have a fixed quantity, and you're speaking of the entire set. So you might say "Two of my dream cars are a BMW and a Mercedes" if you mean that you also dream about a Porsche and a Jaguar. However, if the first two are the only ones you dream about, you'd simply say "I have two dream cars, a BMW and a Mercedes." Therefore, when answering the hypothetical question, the elliptical form to use would be simply "I have two," not "I have two of them."

As for "there are two": it's better than either of the "of them" alternatives, and there isn't all that much to choose between it and "I have two."