It's Tere not Tera because in "Raaj tere ghar men rahataa hai", the 'men' (and any other postposition- I think that's what they're called- 'in, on, of, towards,' etc.) makes the 'tera ghar' go into the oblique. Ghar, itself, never changes, so here, the tera alone turns into tere.
To the course creators/maintainers: since you are aware that the "oblique case" involves non-intuitive grammatical trickery, why don't you have a separate, dedicated lesson for it? Entering the lesson called "Family", I expected to learn about family related vocabulary; I got this sentence as one of the first, and it took me a while to figure out its structure properly. I suggest you create a lesson with a title like "Unexpected grammar: Oblique case" and teach about the issue there. Also, if you can, at least in the basic (Level 0/5) examples use words that demonstrate the case clearly by changing form: as "घर" in the oblique case looks exactly the same as in the nominative, it is as helpful as using "you/you" to teach the accusative case in English instead of "he/him".
It's as Kateykr described above. You use "tere" here not to imply that it's plural but because it's an oblique case. If you refer to the object directly like "this is your house" you would use "tera". However, in this case it's not just "your house" it's "IN your house"; the preposition "in" makes it an oblique case so you use "tere".
Another example would be "You are my son" - direct case, use "mera". But, if the sentence was "You are with my son", then it is an oblique case so you use "mere".
That's because the oblique case of तेरा is तेरे. since raj lives IN your home, everything that the preposition "in" refers to (that is "your home") have to take the oblique case.
Even घर is in oblique case here, but since it's the same as the regular case, that is easy to forget.