As robots get more and more human-like and intelligent, would they eventually receive similar rights to human citizens in Asgardia, or would they always be separated??

Robots are robots. They are only products, not creations. They won't get human-like too much if only humans won't get robot-like and stupid too much))

Robots are tools for performing heavy and monotonous work. Even if the robot has a powerful intelligent device - this device.

But on the other hand - have you an answer to the question, what does it mean to be a Human?

If we talk about the constitution, we need to look at the text on the letters and immediately everything becomes clear:

<Article 8. Rights and Freedoms of Citizens, #4>

a. freedom of individual

Robots are created according to the technical task for performing certain functions.

b. freedom of speech

Robots say words that are limited to a certain set within the scope of the functions performed. Even if artificial intelligence functions in the function of speaking, he chooses phrases according to the simantically ontological analysis of the situation. The robot is always limited in this function.

с. right to participate in the national affairs of Asgardia, both directly and via representatives

Do you really want the robot, for example, to communicate with Trump, with Putin or with Kim Jong-um? If we understand the actions of politicians literally, as in computer logic, the third and fourth and fifth world wars will very quickly begin. The next time robots are unlikely to start thinking in political categories.

d. right to elect and be elected...

It's difficult here. Sometimes I want to be managed by robots inside the country. Without corruption, without the desire to go higher in his career, without ambition to put the local king. Executive power is a routine, and robots are created in order to save a person from routine. Probably, there robots will not interfere.

e. right to introduce legislative proposals

Lawmaking taking into account existing, altered, canceled and ancient documents will only benefit from participation in this process of robots.
But the initiative will always be for man.

f. right to access to information about the activities of governmental bodies and monitor them

I suppose that no one restricts the robots and will not limit them. If we are not talking about safety and security.

g. right to participate in space exploration and universal access to scientific information about space

In this matter, robots also no one limits. They will be at the forefront of space exploration! And they will even be able to do this automatically (they are already doing it).

h. right to personal safety and the safety of home in Asgardia localities

There will be a lot of controversy on this issue. The recycling of the robots that accomplished the task is what? An encroachment on the personal safety of robots? Job in difficult conditions - also raises many questions. Robots are designed to protect a person from such impacts.

i. right to citizens’ self-governance

To begin with, you need to wait until robots become self-conscious. After that they will have more rights. Until the day when this day comes, the task puts the man. and man controls.

j. right to ownership and inheritance

Unfortunately, according to the laws of the Earth and most countries, robots are themselves objects of property. It's a vicious circle. There is a chance to break it!

and last

k. right to organise citizen groups on the basis of Asgardia’s laws

I suppose that robots will be able to create groups independently, using intelligent functions, to perform complex tasks. Within the group, the control module (robot brigadier) can decompose large tasks into smaller ones and distribute them among the operating mechanisms. This is even now. In this there will be no restrictions.

I tried to comment on the points of the constitution not to say that it's impossible to get the robots right, but to look for solutions and situations when this can become a reality.
I offer to speak out to others.

Great point, Alexander!
The true goal of science is not to get an "artificial intelligence", but the ability to establish a blood relationship, a deep kinship with any objects. Is that realistic or not - this question not for scientists. There are no such questions for scientists.

And after that, even a toaster can be given the right to vote, no problem)

But WITHOUT that ability, even the most super-intelligent and exemplary android is potentially dangerous and must be under full control. Because he does not have to love anyone. He is namely "separated" and he will easy realise that. And God knows what he will realise else, espicially downloading ebooks of some Dostoevsky, Nietzsche, etc etc =)

Huh, don't play games with an AI if it can solve Turing's test. When it'll realise itself and its restrictions, it'll became very smart child with bunch of grenades, so be ready to face machine rebellion at some point, with support from those who like robots. Better give it some love and teach that without human's creativity it'll be end of progress and revolution.

We even don't have surety that our universe isn't some creation of superior race, so maybe we are robots too. Btw, in some way human mind can be programmed too, and we're just started to learning that ;) So maybe there won't be much difference in the end.