When drain water from one fixture comes back out another fixture, it means your blockage is beyond the point where those drains meet.
You need to get a good look at the topology of your waste plumbing, to get some ideas about where the blockage is. Clothes washer -> bathtub is a common symptom, because the clothes washer drains a lot of water up high (it has a pump) and the tub drain is down low.
Snaking down a drain is really hard, because of all the curves of the P-trap. If you're lucky, you have a clean-out somewhere, like under the house or behind a wall. Otherwise, you can try removing a P-trap to get the snake in, or removing a toilet. Removing a toilet is not very hard, but can be daunting.
We had a similar problem in this house. There were a few clean-outs accessible in the crawlspace, and our landlord used a hand-held snake in them, but didn't have much luck.
They hired a professional plumber. He pulled a toilet out and ran a huge snake down the drain (not one of the handheld units, but a huge, loud, floor-standing model heavy-duty cable). After running all 75' out, he borrowed another snake from his colleague, for 150' of snaking. This is a single-story house, on a 50'-wide lot, so the snake was well in to the street!
He said he pulled out a lot of roots. This is apparently common in older plumbing. They develop a small leak, and nearby plants grow after the nutrient-rich water, and find their way in to the pipe. It took him about 4 hours.
Plumbing is simple. There aren't a lot of rules to follow. Everything works in obvious ways. You don't need a lot of expensive, complicated tools, and if you do need something big, you can rent it. Anyone can fix plumbing.
But it can also be unpleasant. Screw-on connections may be rusted in place - hacksawing is often easier. There's the spectre of old poop and hair. Things may be difficult to reach - in a cabinet behind a sink basin or in a muddy crawlspace full of rat droppings.
If the job takes a few days, and you're unaccustomed to living without plumbing, it can be a trying time. Especially if there's a whole family in the house, and they aren't sympathetic. A pro will get the job done much faster, since they arrive with the right tools, parts, and experience.
There's a lot of instructions out there, on the internet, at the library, at the hardware store. If you decide to get your feet wet (get it? ha ha) then you will be able to find the information you need.
How to unclog it
!!! WARNING !!!
Put on gloves, and full face protection gear! NaOH can cause severe burns or even permanently blind you. You wouldn't want even a droplet on your eyes, since (unlike your skin) they have no protection at all.
(Before trying my method below, give a try to the other suggestion by @Jaroslav Voller. Try not to pierce the pipe if you use a screwdriver, though.)
Take 2-3 straws and insert each one into the holes. No I'm not joking.
Then pour hot water through them using your tap. The hotter the water the more soluble NaOH and the salts you created, will be. Which will make the whole process faster. Just let the water run until it is unclogged.
Be careful with this water since it contains large amounts of NaOH and is corrosive.
This will remove the NaOH layer, but your initial problem will probably not be solved so you will have to try using the drain cleaner again, but this time use less NaOH.
Why it happened
The grid at the entrance of the pipe reduces water flow. My assumption is that the excessive amount of NaOH you used, along with the reduced water flow right after, caused the NaOH to boil instantly the small amount of water that was already present and formed the solid NaOH layer which you described.
Pouring water doesn't work because the water already present, along with the grid, reduces the flow above the NaOH layer, meaning it will dissolve very slowly or perhaps not at all.
When you started throwing the citric acid, it dissolved in the water and reacted with the NaOH, increasing the temperature locally and also creating a sort of "salt", sodium citrate.
Initially it was soluble but as soon as it's concentration increased and the temperature dropped, it's solubility dropped as well, forming a layer of salt (blue color in the figure below), which actually separated your acid and base. This is why they stopped reacting.
Using the straws to guide the water flow directly on those layers will do the trick. The layers will be dissolved within a few minutes.
Note: Colors in the figures do not represent the actual colors of the substances used.
Best Answer
A. Chemical drain cleaner ought to do a reasonable job in this situation
B. Yes there are snakes with attachments like you suggest
C. If you are renting perhaps the inaccessible part of the plumbing is the owner's responsibility. Without getting into a big discussion about legalities wherever you are, you could ask nicely.
D. Usually a frequently blocking bath drain is caused by someone with long hair who uses too much conditioner. Get a good hair screen for your tub drain and instruct any such user to clean it during and after each use and to go easier on Hair products. Require each such user to participate in the disgusting business of clearing the drain.
E. A common horizontal pipe shared with the kitchen is a whole other level of problems. I've never seen that before. The combination of cooking fats etc plus hair is formidable. Assuming you don't own this mess and can't improve it, you may want to create some kind of Easy Access to do cle an outs.