Learn English – I’d like to know whether the usage of word “concerned” in the following sentences are correct or not

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)

Any of those sentences could be used and seem completely natural in different contexts. You choose each among those expressions based on what you want to communicate.

Are you talking about one chair or many chairs? That (obviously) determines whether you choose the plural or not. If I asked about multiple spoons, but you answered me with a singular response, it would seem odd. For example, if I said "Would you get me two spoons from the drawer please?" and you responded "There's no spoon here." If I ask about a singular and you responded with a plural it wouldn't be as odd. If I say "Go tell Joe he has a phone call." you could say "There are no Joes in this room - I don't know who you're talking about." even though it's more likely you would say "Joe isn't here."

Are you talking about one specific chair or just any chair? That determines whether you say "there is no chair" or "there isn't a chair". This can be tricky because it really depends on what is in your mind when you say it, and not necessarily anything that has been said earlier. I could say "When you get there, sit in the chair." and if there are only couches when you get there, you might tell me "There is no chair here." (the specific chair that you were going to sit in isn't there) or "There aren't any chairs here" (you looked for any chair to sit in and there were none available).

In the movie the Matrix, there is a scene where a boy is bending a spoon with his mind and he says that to do that first you must realize there is no spoon. He is speaking about a specific spoon. If I asked you to get me a spoon from a drawer that had no spoons in it, you could say "there isn't a spoon in here (for me to give you)." or "There aren't any spoons in here, just forks." or "There's no spoon in here (the specific spoon you want isn't here)." All of those statements are fine and sound natural.