The Imperium is often hell in general, but there is no specific focus against LGBT folk. For all it takes the aesthetics of Catholicism, the Imperial Cult is a human-focused religion and does not seem to proscribe sexual or gender categories, at least no more than it proscribes anything else we might consider a religious issue.
The Imperium as a whole is divided into disparate societies and cultures, all of whom are likely to have different opinions on these things. The culture of "the Imperium" is generally in reference to the cosmopolitan galactic culture of the Imperium's spacer and planetary capital populations. These folk are often, by Imperial standards, educated and more aware of the horrifying alien threats that are faced on all sides. Such people are unlikely to view gender or sexuality in an ideological light, as they are at a level of civilization where any concerns about that don't have a real effect on anything.
Planetary cultures are likely to respond based on local prejudices or lack thereof. Feral or feudal worlds are a complete wash, much as it was in our own history. Some places, like hive factory habs, probably put drugs in the water to make people breed or not breed, depending upon what the management requires. Civilized or paradise worlds are probably the most likely to have anything resembling modern tolerance of multiple lifestyles, in LGBT or in anything else.
However, what the Imperium definitely has is a dirth of culture, which is a large part of why it's so horrible (the other part is your painful death). As such, even conceiving of how a particular social group is treated is something that's just not possible for many worlds.