Conducting research on homosexuality would indirectly means that we actually consider it a disease.
The logic of this statement is erroneous, the assumption that the ultimate logical consequence of inquiry (research) equals questioning the judgement of Allah is to impound a rule that criminalizes the inalienable right of thoughts. Islam unlike Christianity does not punish again "thought crimes."
Islam refers to the niyyah (intention) of a person when in relation to actions, the person can be doing a research case to be only trying to understand the nature of homosexuality, not to be "considering" it a disease. So the "indirectly" part is based on a assumption which can be disregarded, as it is based on ones niyyah, not thoughts.
i.e. did Allah punish those who were in essence ill or having some sort of disease?
This brings up something interesting, how do you know the homosexuality of the past, is the homosexuality of today. There are cases known of when heterosexual people sleep with the same sex, yet still identify heterosexual, and live heterosexual lifestyles (i.e. marrying the opposite sex, scientists notice this, and some assume it to be just sexual release, as men mostly have the same libido levels).
Words of the Quran about this and the action of Allah on them, seem to, establish a point that homosexuality is actually a voluntary and rebellious choice and which can't be considered as a disease.
The sentence above is speculation, while I give you that the Qur'an does seem to establish a point that the people of Lut did commit "homosexuality" voluntarily, and as a rebellious choice, there is no way to know if the homosexuality of the past referenced is the homosexuality of today classified by psychological associations. As well as with reference to the above answer/paragraph, where heterosexual people sleep with the same sex, yet still identify heterosexual, and live heterosexual lifestyles it could be to this that the Qur'an refers to.
Plus, based on the Islamic belief is it incumbent on a Muslim to consider the words of homosexuality apologists and who state that those sexual orientations are just natural and that's how nature made them, as blatant lies? And never agree with the results set forth by researchers that it is a natural disposition or influence taken from the surrounding?
No, as I have reiterated many times in this answer, the Islamic belief does not make it incumbent one a Muslim to forbid the agreement by researchers that it is a natural disposition or influence taken from the surrounding. As the basis that the homosexuality described in the people of Lut story, is the same as homosexuality defined by today's society is a assumption, and a weak one at that.
Is it permissible to conduct research on homosexuality, it's cause and try to formulate psychologically cure for it?
Socially I would assume many people would classify your actions as hate-related, however under Islamic sharia there is no issue to conduct research on homosexuality, and try to formulate psychologically cure for it.