This seems to be a common misconception among many Jews also, so I'll try to address it as clearly as possible:
There are 2 separate parts to your question - 1. Why do we not make requests of G-d on Shabbos, and 2. Does G-d do work on Shabbos (and when does he consider it Shabbos)?
In regards to the first question, the reason why we don't make requests of G-d has nothing to do with whether or not he is "resting", but rather, as the Talmud Yerushali says in 'שבת פרק טו הלכה ג, that we do not make requests about our personal needs, with the exception of certain prayers (eg. אלוקי נצור וברכת המזון) whose language is set and identical to that of the rest of the week, is because making requests shows that one is lacking something, and has needs, which is a lack in מנוחה (roughly translated as rest, but includes a completeness).
This being said (that our lack of making requests of G-d on Shabbos has nothing to do with him, rather with us), G-d does, in fact, also "rest" on Shabbos (in regards to question 2). However, as Rebbi told the Emperor Antoninus in meseches Shabbos, while it does rain etc. on Shabbos, that is not G-d doing acts of creation (the true definition of melacha, "creative work"), but rather him moving things around in his own domain. There is no question of "somewhere in the world it is not Shabbos", as G-d refrains from doing melacha in each place when it is Shabbos there.