The Shulhan Aruch (Orah Haim 4:10) instructs that when performing Netilat Yadayim, one should fill the vessel with one’s right hand, pass it to the left hand and use the left hand to pour the water on to the right hand first. The Mishna Berura (O.H. 4:22) explains that the right hand represents the attribute of kindness (Hesed) while the left hand represents that of strict judgment (Din). By washing in this manner, one humbles the Din, as it were, before the Hesed. TheBen Ish Hai (Parashat Toledot, § 1) even says that one should not pass the vessel directly from the left hand to the right, as this demonstrates a certain level of honor to the left hand. Therefore, he writes that one may pass directly from the right to the left, but in the opposite direction one should place the vessel down with the left hand and then pick it up with the right hand.
The Shulhan Aruch (ibid:11) goes on to say that one should not have one’s hands washed by someone who did not do Netilat Yadayim in the morning. This circumstance may arise in the case of a Levi washing the hands of a Kohen, or an elderly person who is having their hands washed by someone else, for example. The rationale is that one who did not wash in the morning still has the Ruah Ra’a and thus has a degree of impurity, such that trying to wash and purify someone else will be ineffective. Rabbi Haim Palagi (Lev Haim, vol. I, § 68) goes further and says that a non-Jew should not wash the hands of a Jew, even though the former is not affected by Ruah Ra’a. Nonetheless, Rabbi Avraham Buchach (Eshel Avraham) says that the washing of a Jew by a non-Jew is effective Thus, if one is in a hospital or is incapacitated, one should try to have a Jew wash one’s hand for Netilat Yadayim, but if not possible, one may rely on a non-Jew
Summary: When starting Netilat Yadayim, one should take the vessel of water with one’s right hand, pass it to the left hand and wash the right hand first. If one cannot do Netilat Yadayim for oneself, one should not rely on someone who did not do Netilat Yadayim in the morning, and preferably not a non-Jew. If necessary, one may rely on a non-Jew.