"I wash my clothes with soap."
Pseudo-correct, but sounds very unnatural. It equates to, say: "I wash with soap my clothes." The を indicates the direct object with regards to the verb, so the thing being washed needs to precede it, other descriptors and their particles can be elsewhere in the sentence but every particle serves to influence what precedes it in different ways and in Japanese you usually end a sentence with the verb as well, so mind your groupings.
Isn't it the other way around? The further to the right in the sentence the greater the emphasis. This is the equivalent of saying "I was my clothes with SOAP" as opposed to washing them with some other substance. That seems to accord to the intent of the English sentence.
I am more curious why 洗濯します is not an acceptable verb here.
You are, but Japanese is HIGHLY contextual. There is usually very little need to specify to such an extent as certain liberties to meaning can be assumed. You don't normally go about washing other people's clothes, so the statement that you do it a certain way generally applies to you and your possessions, even your households' i.e. 私と私の家族は石鹸で服を洗います
What I believe Sora is saying is that the kanji 鹸 can be difficult even among Japanese people because it's not a regular-use kanji (it's not one of the 2,136 jōyō list kanji).
I needed to buy a bar of soap when I was on holiday in Japan and on it was written 石けん instead. I've found a picture of it:
* edit * Wow, that picture came out big... xD