"Let us know when he arrives home" would be the same plus "o"?
It's the same and o would be optional.
In case "o" is used, a comma is necessary: O, eve ulaş...
Would "Inform us ..." for "... bize haber ver" also be acceptable?
It is fine and was just forgotten. It does sound a bit robotic in a sentence like this, but it is totally acceptable. :)
How does the locative ending on ulaştığında (the "-da") become "when"? I would assume that it'd translate as something like "at".
In the tips and notes to this lesson it is explained. Using locative many times changes the meaning to "when" and using Applative many times changes the meaning to "because of".
I understand it as something like "Let us know of his (being) at home", what with "to be" being often implicit in Turkish
Sorry, but what is the difference to the sentence ''eve gelince bize haber ver''??
I think that there is no difference in meaning. In Tips and Notes they wrote that the meaning is very similar. However, I am learning too.
Yes no difference both of two