Your first two sentences are fine, and sound natural. The third one is correct, but I think that most native speakers (rightly or wrongly) would instinctively use a gerund : '..doing the washing up.' I saw this post yesterday, but didn't answer because I wasn't convinced by this rule, and I also struggled to find examples other than with 'and' or 'or'.

I've been mulling this question over a bit more, and I think I know what's wrong with this 'rule'. I think the problem is that it's not really about bare infinitives at all. I think it's a more general point than this. It's about conjunctions, and the fact that you can 'carry over' the sense to the second part of the sentence, and just use the main verb in the same form as the corresponding verb in the first part, with no need to repeat the auxiliary verb or modal. Here are some examples: He can dance and sing. He has danced and sung He likes dancing and singing. He can't dance or sing He hasn't danced or sung He doesn't like dancing or singing. He'd rather sing than dance. He'd rather have sung than danced He'd rather be dancing than singing. You only follow 'and' 'or' and 'than' with the bare infinitive if there's a bare infinitive before the conjunction. If there's another verb form (such as a gerund or participle) before the conjunction, that's the form that is repeated after the conjunction. So it's not much of a rule! Like you, the only examples I can think of with 'but' and 'except' are along the lines of 'anything but..' and 'everything except', but they don't follow the rules either. It's possible that we've both completely missed the point here and someone else will come up with a brilliant way of explaining it, but I doubt it.

The other teachers have made good points, so I’ll just add some additional insights: As far as what’s wrong with number 3: “Susanne is always willing to help me with everything except do the washing-up.” The reason I’d use the gerund (“doing the washing up”) instead of the bare infinitive is that it is governed by “with” — to help me with everything except (with) (doing) the washing-up. Note that you could easily omit “doing” since it is essentially an “empty” word used to turn the noun phrase “the washing up” into a verb — if you want to use a noun, “to do” is superfluous. So I’d say: “Susanne is always willing to help me with everything except the washing-up.” As far as the general use of bare infinitives after conjunctions, I’d say it’s simply a matter of not needing to repeat the particle “to.” For example: I like to cook, travel, and take long walks on the beach. It would probably not be a mistake to repeat the “to” in my particular example, but usually we don’t repeat it. This is because, contrary to the official grammatical analysis with the particle “to” forming part of the infinitive, native speakers usually analyze “to” (subconsciously) as being part of the verb before it. This is why we have spoken forms such as “usta,” “hafta,” “gonna” “wanna,” etc. While these forms are forbidden in the written language, it’s my understanding that native speakers learn the reduced forms first, and only learn the official analysis when they get to school.

You may have not received an answer because the grammar rule you state is wrong. I'd say it can apply to the word "and" (but often "and" is followed by "to verb".) For the others, I can't think of examples that don't need the "to" or need to be followed by a "ing" form.

Additional Details: The section ‘Bare Infinitives’ (as below) is extracted (with given examples excluded) from the coursebook I mentioned. The bare infinitive is the infinitive without ‘to’. We use the bare infinitive: 1) after the auxiliary verb ‘do’ and modal verbs ‘can’, ‘could’, ‘may’, ‘might’, ‘must’, ‘have to’, ‘ought to’, ‘shall’, ‘should’, ‘will’ and ‘would’; 2) after ‘let’ and ‘make’; 3) after verbs of perception: feel, hear, see and watch; 4) after ‘would rather’, ‘rather than’, ‘had better’ and ‘why not’; and 5) for the second infinitive when two infinitives are joined by ‘and’, ‘or’, ‘except’, ‘but’, ‘than’, ‘as’ or ‘like’.