"Then, I will do the laundry."
Simple present and simple future tense are the same thing in Japanese as far as I know. This is why something like 「パンを食べます。」is either "I eat bread," or "I will eat bread," depending on context while 「パンを食べています」 is strictly "I am eating bread."
I've also just learned that it's a little more complicated than that. Consider 決める（きめる）which means "to decide." Something like 決めている is most accurately "it has been decided" as opposed to "it is being decided." This has to do with the difference between state-based and action-based verbs, and I'm too new to this to always know the difference. That should be enough of a bread-crumb trail to spawn independent research, though. The information in this paragraph comes from teachers and natives from the 日本語と英語 Discord Server found at https://discord.gg/0eIsYvFQul270V1L
I'm not a native, so take this with as much salt as needed. Flavor to taste.
If you look at a JA-EN dictionary, you'll notice a lot of action-ish nouns (like doing laundry) are noted as being "Suru verbs". This indicates that it's perfectly okay to append する or します (which is the polite version of する) to change it from a verb-ish noun into an actual verb.
As far as I know, it's also perfectly valid to include the を, but doing so blocks you from using the を to indicate a target of that verb (So both 勉強(べんきょう)をします and 勉強します are fine, but if you have to use 日本語を勉強します since the を is already used to mark Japanese as the thing you're studying)
Warning: I'm not a native.
In this case, 「せんたくします」("I do the laundry/I will do the laundry") is the present/future tense of 「せんたくする」("To do the laundry"), and 「せんたくしています」("I am doing the laundry") is the present progressive tense. In general, して is considered the ～て form of する, and the ～て form is something worthy of a good amount of directed, independent study.
From what I've gathered the を is optional in so-called suru verbs, but if you use it, you won't be able to use を again to mark a direct object. Since there is no direct object ("do the laundry" counts as a full verb, "laundry" isn't just what the verb "do" is directed at) in this sentence though, it's totally okay.
That's not the same meaning.
それから : それ = that thing (near the listener, or that the listener has in mind), から = from; so the meaning is something as "after the thing that is known has been done"; it places the action of 洗濯する after some other thing.
それなら is a conditional/hypothetical. それ = that thing; なら = conditional form of だ/です ; so the meaning is something like "if that thing is; in the case that thing is";
それ can refer to something in the previously said sentence, yes; but それから means that 洗濯する will occur after what has been told in the previous sentence, while それなら means 洗濯する will occur if what has been told in the previous sentence happens/is true.
English "then" has both meanings, but Japanese uses different expressions.
Same reason it's been omitted for all of the lessons leading up to now for the most part. They're leaving the subject contextual unless they need to specify it, much as is done in spoken Japanese. The only time to use 私 is when it's unclear from context that you're talking about yourself. I keep running into natives that tell me it's condescending to use it all of the time when it's otherwise obvious.
I hope this helps.
私 is often omitted in Japanese sentences. It would still be correct if the sentence was "それから, 私はせんたくします." It would also be more precise, because as another poster said, the original sentence can also be translated as "THEY will do laundry." or "HE/SHE/YOU will do laundry."