Since many poskim hold that not using electricty on Shabbat is just a Minhag (Rav SZ. Aurbech), would they hold that it is 100% ok to ask a non-Jew to do things with electricity for you?
R' Auerbach OBM held that refraining using electricity per se is a minhag. However, almost all electric devices involve one or more melochos, which is why it is a minhag to refrain from electricity use, since unless you are an electrical engineer AND a rav, it is unlikely that you would be able to determine what exactly is permissible and not permissible about a given electric device. For example:
- A computer almost certainly makes a record, falling under the melocho of kosaiv mederabanon, since it's not the "normal" way of writing.
- Anything that beeps or makes noise (stereo?) falls under the prohibition of zilzul shabbos.
- An incandescent lightbulb becomes hot and transgresses... well, there's a machloches.
But what about very simple devices, such as florescent electric lights? The Igros Moshe, for example, held that one would be permitted to turn a light off on Shabbos, but that one should refrain, because of minhag yisroel. Obviously in pressing circumstances, one might be able to do that, even without amira l'akum.
Because of the complexities involved, if you hold like R' Auerbach, you would have to consult a rav about each and every individual device and possibly each case if you want to use amira l'akum to operate electrical devices on shabbos, since other prohibitions are almost certainly involved.