The landlord has a legal obligation to address repairs. That's what your rent covers.
Instead of calling him, write a letter that says you've spoken to him X times over Y months about it, and if it's not resolved by him per the lease agreement within 14 days, you will contact your own plumber and deduct the cost of repairs from your rent. I think you can reference that you have called him and feel that it's no longer productive to do that because he screams at you, blames it on cats, and so on. Make your statements extremely matter-of-fact and do not exaggerate. Make it business-like and professional. You have a contractual relationship with the landlord - it's your lease. You are not at his mercy. Do NOT reference your pregnancy - it's irrelevant to the repair issue and just gives him an excuse to say that you're hormonal. (It may be why you don't want to listen to him, but it's not why your sink is bubbling.
I do think there may be value in contacting the city sewer department or the water department (if you are hooked up to sewer). Otherwise, if you're not on sewer but are on septic, I'd say your septic tank might need to be pumped if it can't handle the amount of water draining from the machine so fast, which would also be a landlord expense. However, it's curious that it doesn't happen when you run the shower - so that would point to a pipe problem in the area of the house where this occurs. In any case, you're entitled to repairs.
Take a picture of the suds bubbling up through the sink. Attach the photo to your letter (just print it on regular computer paper - don't bother getting a fancy developed photo). If you can take a video, do that too. If you are emailing the landlord vs. sending a hard copy, attach the video in addition to the photo.
If you can get someone from the water/sewer department there, then run a load of wash when they are present. They can observe the smell as well. Ask them outright if this is related at all to sewage - that's a health hazard. I think rotten eggs is a sulphur smell, right? So maybe you have sulphur in your water? Are you on a well or on city water? Those are all questions to ask the water department. You may also be entitled to free water testing from the city. Ask.
You have every right to deduct from the rent for your expenses, and if you have to go to a laundromat to avoid using the washer hookup provided. Do not be afraid of eviction - it's very very difficult to evict a tenant anyway (even one who's in the wrong), and he certainly can't do it over a disputed/neglected repair.
Good luck!