I think you mean authoritarian state. Fascism is a specific right wing thing.
"All within the state, nothing outside the state, nothing against the state."
-- Benito Mussolini
That is anything but right wing.
Fundamentally, the left is about collectivism and the right is about individualism. The difference is whether you think the state is a provider or a parasite. It's about wanting the government to take care of you or get out of your way. It's about wanting the government to make everybody have equal stuff or an equal opportunity.
No, its not. That's pretty much the libertarian-authoritarian axis, which is largely orthogonal to the left-right axis. Libertarian socialism is not only a thing, it's an older thing than modern, pro-capitalist right-libertarianism. (It's also older than Leninism.)
Perhaps you are calibrated to a non-US definition. If so, you should qualify it as something like "UK right" or "UK left" or whatever.
Relative to the center in the US they are on the left and also, if not on the authoritarian side, at least not on the libertarian side either.
> and you don't think Rand Paul and Clarence Thomas are on the right.
Paul is, rhetorically at least, a right-libertarian. Thomas is a partisan Republican who is quiet enough on ideological issues that I wouldn't try to place him other than somewhere vaguely on the right.
> Perhaps you are calibrated to a non-US definition.
No, actually, my degree is focussed on the US political space, not that it really matters (the US center is way to the right of the developed world, such that what is by broader standards a center-right faction has recently been dominant in the left-most of the two major parties, but that just affects position on that axis, not what the axes are; it's also true that US politics seems unidimensional due to the two party system, so all variations often get characterized as left-right even when they aren't and only weakly, if at all, correlate with left-right variation.)