Probably because you'd use the phrase a lot when speaking with Japanese waiters at restaurants. After all, if your pronunciation is really authentic when pointing to menu items, saying "This one please," they may ask you to elaborate on how you'd like it prepared, not realizing you are not fluent.

Because this lesson introduces the negative form of verbs, I think.

When the verb of a japanese sentence is negative, the が can be replaced by は.

はなせません　is I cant speak. Whereas はなしません is more like I dont speak.

First off, tea is おちゃ, not いちゃ.

Now onto your question - のむ (to drink) is a transitive verb (also called an "action verb"). That means it can have a direct object - the object to which you're performing the verb action (drinking, in this case). Direct objects are marked with を.

はなせる (to be able to speak) is an intransitive verb. It is not an action verb, merely a passive statement of one's ability. That means that it cannot have a direct object, and thus cannot go with the を particle.

You might be getting it confused with はなす (which is where はなします comes from), which is the transitive verb for "to speak." That is an action verb which can have a direct object and use the を particle. はなせる (which is where はなせます comes from) is a different, intransitive form of はなす though, called the "potential form." It is used to describe your ability/potential to do the base verb. These are always intransitive.

To go back to your tea drinking example, if you were saying that you cannot drink tea, rather than just that you don't drink it, it would be おちゃはのめません. Notice that since we're using のめる (intransitive potential form) instead of のむ (transitive action form), we don't use を anymore.

私達は日本語話せません。word 達(たち) Makes word 私(わたし) into plural form

“I” would be the default implied subject if there is no other to be inferred from context. Given the right context I guess “you” is technically possible, though I can’t really think of a situation where I would inform somebody that they can’t speak Japanese – in Japanese! But “he” or “she” for example are easily possible if we’re talking about some other person in context.

は does not mark the subject but the topic, the main thing that the sentence (and potentially following ones as well) are talking about. The topic can be identical to the subject (in which case there is no further marking on it) but it doesn’t have to. This is where the が particle comes into play: It marks a subject which is not the topic.

はなせる is derived from the verb はなす “to speak”. This form talks about whether or not the verb can be done – and it is almost always used intransitively (i.e. meaning it only takes a subject but no object). If it helps, you can はなせる as having passive element: “can be spoken”. So the construction 日本語ははなせません can be roughly imitated in English as “Japanese cannot be spoken [by me].”

Please have a quick scan if your question was answered before asking :) はなせる is a derived version of the verb はなす “to speak” which expresses possibility: “can speak”. However, this verb form includes a passive meaning, meaning it’s literally more along the lines of “can be spoken”. Therefore, the language is actually the subject of はなせる not the object.

話しません is the plain form "Do not speak", you need the potential form 話せません "Cannot speak"

I'm not sure what you mean, can you be more specific?

That is correct.
The answer to this question is 日本語は話せません (kanji) - にほんごははなせません (hiragana) - nihongo wa hanasemasen
(japanese language) (cannot speak)

はなせ. Although in this case 話せ is not a complete word, just the stem of the verb 話せる(はなせる) “to be able to speak”, itself a derived form of 話す(はなす) “to speak”.

This form is sometimes called the “potential” because it basically takes a verb “to do x” and turns it into a new verb with the meaning “can be x-ed”. So 話す “to speak” becomes 話せる “can be spoken”. Confer this excellent description for details.

Note particularly that the potential form always includes an inherent passive: “to speak” strictly speaking becomes “can be spoken”, not “can speak”. So the object of the original verb becomes the subject of the potential (though it’s often made the topic with は). This is why you say 日本語は話せる (literally: As for Japanese, it can be spoken), not *日本語を話せる.

語 is “go” as in 日本語; you probably meant 話. That’s kun'yomi. The on'yomi is “wa” (as in denwa 電話 “telephone” for example).