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, room means a space with specific dimensions. As Damkerng T. noted, it implies that the space is sufficient for a certain purpose.
Make room for me!
Means: Clear a space of sufficient dimensions for me to sit on.
Go mark down a 5" by 5" space in our yard, please.
In this example space is being used to refer to a space of geometrical dimensions.
You could not use the word room here, which is defined as an extent of space occupied by or sufficient or available for something.
To further clarify, take a look at the following example including both the terms space and room:
Please mark down a space^ in our yard with room^^ for our car.
^ no dimensions specified, so far it could mean a space of any size
^^ room for our car clarifies that the the space must be big enough to fit an object of size X.
Also, you can use the word "spot" (a particular place, area, or part) to refer to a space that's of sufficient dimensions for something.
I need a spot for my car.
Spot implies that the space needs to be big enough to fit a certain object as well.
Best Answer
To my (American) ear, "room"'s connotations are slightly more 2-dimensional, and "space"'s connotations are slightly more 3-dimensional.
Thus, if you have so much clutter that you don't have "elbow room", or cannot find any more floor area for activities, "You have no room in your house."
If you have so much clutter that you cannot find a place (even up high, or down low, or in a closet or cabinet) to put anything, "You have no space in your house."
It is easy to imagine someone who has unused space in their house, who does not have any more room for new activities (or for people to move in). It requires a dedicated "hoarder" to use up all of the space in a house.