Which is correct grammatically?
Sein Haus und Garten ist schön.
Sein Haus und Garten sind schön.
I tend to think the former is correct by taking Sein Haus und Garten as a singular subject. Is this right?
The latter is correct. Haus und Garten refer to two tings so that is plural; as simple as that.
Note: The sentence is shortened. You could extend it to
There is an old german song that goes...
even there it´s wrong, because it must be
in colloquial language you actually hear that a lot. So if you make a mistake when talking to somebody, most german people won´t even notice. But it´s still not right to use the singular here.
Like marstato says, you're talking about more than one object, so the plural applies:
OTOH if you say
then "Haus mit Garten" becomes a single object, and then the singular applies.
Note: a common error in many languages is to use the plural form also for "a number" or "a lot":
(Can somebody make this an example in German?) This is wrong. There may be many things (plural), but "a number" is singular. The correct form is
You don't say
either, do you?