Nayaly. Así como en español, en inglés, los sujetos "nadie" y "alguien" se conjugan con la tercera persona singular. Por ejemplo: Nadie sabe, alguien sabe, él sabe. Nadie lava, alguien lava, él lava. En inglés es lo mismo, simplemente piensa en cómo conjugarías el verbo con el pronombre "he" y esa es la conjugación que debes usar. Nobody knows, somebody knows, he knows. Nobody washes, somebody washes, he washes. Lo mismo va para "no one" y "someone". Espero que eso responda tu pregunta.

Someone who washes his/her own clothes.

Very good. The only correction would be to say "washes" instead of "wash." The verb "wash" is describing what someone (not you) are doing to the item in the present (now). Since there is more than one of the item (the clothing) and this is happening in the present, "wash" should instead be "washes." There are some exceptions though. If you say "He washes his car," the verb "wash" is in the same form as the sentence you just gave, because you are describing that person. I hope this makes sense. If you have any questions please ask!

Close, but you would say "someone who washes their clothes" and then you would add something to the beginning or end, like "Someone who washes their clothes is cool." or "He is someone who washes their clothes." Hope I could help and send me a message if you have any further doubts.

NO, IT IS NOT BECAUSE THE WORD "SOMEONE" IS SINGULAR BUT NOT PLURAL.