What rulings and opinions are there in regards to speaking HaShem's Name in its short form, namely [pun intended] י, qamatz, and ה?
Yes and no. יה has the same set of restrictions as other names like אלהים, אדני, שדי, and אל. E.g. it may only be said in an appropriate context, and in a clean place and state.
It does not have a prohibition of יהוה, which it is prohibited to pronounce under other than under very special circumstances in the temple.
Indeed, יה can often be heard in prayers, and songs. For example at the end of Ashrey, and in the very title of Kah Ribon.
Note that the restrictions on pronouncing holy names are only relevant when the intention is for the name to have holiness. It is therefore permitted to answer yes to a question in Swedish, even if one pronounces יה identically during prayer.
For sources, see here.