This is a very broad question, even though you tagged it as "netilat-yadayim-washing".
Regarding washing- since we wash as a Gezera that hands have Tuma that the rest of the body doesn't, it would follow that a prosthetic limb would not need to be washed (for Halachic-hand-washing reasons. But possibly yes for hygiene reasons).
Similar to wearing gloves which get you around a washing issue, if there's no water available or in other emergencies. See Kitzur Shulchan Aruch 40:14 - אֲבָל אִם גַּם שָׁמָּה לֹא יִמְצָא מַיִם, אוֹ שֶׁהוּא עִם חֲבוּרָה וּמִתְיָרֵא לְהִפָּרֵד מֵהֶם, וְכֵן מִי שֶׁיֵשׁ לוֹ שְׁאָר אֹנֶס שֶׁאֵינוֹ יָכוֹל לִטּוֹל יָדָיו, יִכְרוֹךְ יָדָיו בַּמַפָּה אוֹ יִלְבַּשׁ בָּתֵּי יָדַיִם (כְּפָפוֹת) [הַאנְדְשׁוּה] וְיֹאכַל כָּךְ.
Regarding a corpse, the Kitzur Shulchan Aruch says at the end of סימן קצז - דין התכריכין והטהרה. ואסור הנאה במת
מֵת, בֵּין גּוֹי בֵּין יִשְֹרָאֵל, וְתַכְרִיכָיו, אֲסוּרִין בַּהֲנָאָה. וְכֵן נוֹיֵי הַמֵּת הַמְחֻבָּרִים לְגוּפּוֹ, כְּגוֹן פֵּאָה נָכְרִית שֶׁהִיא קְשׁוּרָה אוֹ קְלוּעָה בְּתוֹךְ שַׂעֲרוֹתָיו, אֲסוּרִין בַּהֲנָאָה. וְכֵן אִם הָיְתָה לוֹ שֵׁן תּוֹתֶבֶת, תִּקָּבֵר עִמּוֹ. אֲבָל נוֹי שֶׁאֵינוֹ מְחֻבָּר לְגוּפוֹ, מֻתָּר. וְכֵן נוֹי שֶׁאֵינוֹ כְּעֵין גוּפוֹ, כְּגוֹן הַתַּכְשִׁיטִין וְהַבְּגָדִים, מֻתָּרִים בְּכָל עִנְיָן.
Anything connected to a corpse is forbidden for use (and needs to be buried) - so if the prosthetic limb is connected at the time of death, it would be considered part of the body.
However, if the prosthetic limb was disconnected at the time of death it would not be considered as having anything to do with the corpse.
Regarding Tefilin Shel Yad, it would seem obvious that you couldn't put them on a prosthetic limb.
And the list goes on...