Stuff like that could be a part of the linotype taketype collection(s), i've seen this font before, but … no, it's very quirky, i think it is a freefont.

>Is there a term for fonts that have a continuous rule above the type?
I don't think so, this font mimiks Hindi

That's Samarkan/Shalimar.

You're too cool for school. Thank you!

I wouldn't know what to call this style of fonts, but what I can tell you is that they are inspired by Hindi script.