Does the following sentence mean I prefer to go home, or something is just as fine as something else?
I'd just as soon go home.
Does this other sentence mean that either thing is good?
I'd just as soon go home as to go to party.
It can also be used to indicate that both options are negative, for example:
Would you like to go to a party?
I'd just as soon be eaten by wolves!
In this case, the respondent is using a situation which is clearly unpleasant (being eaten by wolves) to indicate that they don't want to go to a party. It should usually be easy to tell this is the intended meaning due to the hyperbole involved.
However, in your example:
I'd just as soon go home as go to a party.
The speaker may be indicating that whilst they don't necessarily want to go home, their preference (or lack of preference) for this is near-equal to that for going to a party. In this case, you would have to look at the wider context to understand the intended meaning.
I don't think I've ever heard it used differently to this (implying negativity), but this might just be unique to the British English I'm familiar with.
Does the following sentence mean I prefer to go home, or something is just as fine as something else?
Yes, I'd just as soon go home means you prefer going home.
Does this other sentence mean that either thing is good?
Sort of; it means both options are good, but you still prefer going home. It's a kinder and gentler version of number 1 because it acknowledges that you heard the other person's preference to go to the party.
It means that both options are equally desirable (or undesirable).
Do you want to go to the movies?
Sure. Or I'd just as soon stay home.
Both options, going to the movies, and staying home, are equally appealing.
Do you want to go the party?
I'd just as soon be eaten alive by a pack of rabid weasels!
Both options, going to the party, and being eaten by rabid weasels, are equally unappealing.
Abbreviation for “I’d as soon to as not”, but in West Texas is way of accepting generous offer without seeming overly appreciative.