From the sunnah we know that we must not spread contagious diseases:
Allah's Messenger (ﷺ) said, "Plague was a means of torture sent on a group of Israelis (or on some people before you). So if you hear of its spread in a land, don't approach it, and if a plague should appear in a land where you are present, then don't leave that land in order to run away from it (i.e. plague). (Sahih al-Bukhari)
While we were in Medina we heard of plague having broken out in Kufa. 'Ata b. Yasir and others said to me that Allah's Messenger (ﷺ) had said. If you are in a land where it (this scourge) has broken out, don't get out of it, and if you were to know that it had broken (in another land, then don't enter it. I said to him: From whom (did you hear it)? They said: 'Amir b. Sa'd has narrated it. So I came to him. They said that he was not present there. So I met his brother Ibrahim b. Sa'd and asked him. He said: I bear testimony to the fact that Usama narrated it to Sa'd saying: I heard Allah's Messenger (ﷺ) as saying that it is a God-sent punishment from the calamity or from the remnant of the calamity with which people were afflicted before you. So when it is in a land and you are there, don't get out of it, and if (this news reaches you) that it has broken out in a land, then don't enter therein. Habib said: I said to Ibrahim: Did you hear Usama narrating it to Sa'd and he was not denying it. He said: Yes. (Sahih Muslim)
What is the ruling on ill people who have contagious disease. Can they be hindered from entering a mosque? and how or rather who is liable for that?
What about them performing hajj?
I mean compared to the daily prayers which are an "ongoing worship" which must be performed each day, hajj is a worship which must at least be performed once. And some disease at least don't have an effect on your strength so basically one would be healthy enough to perform it.
