Definitely remove the P trap where the two sink drains join, if you are leaving the P trap under the floor.
There is already a P trap in that drain line, and you must not install a second one in series. This was already mentioned at least twice yesterday. This is now the third time you have read this fact in this thread. Never have two P traps in the same drain line. This is now the fourth time you have read this fact. If I write it out again, in a different way, you will have read it five times. Your two sink drains get connected together before their now-common drain pipe goes to a common P trap, which is (I guess) the one under the floor, if you end up keeping that P trap and using that pipe for the two sinks. Hope it makes sense.
To compensate for the large distance that water would fall to the underslab P trap, you can make that drain pipe go like a Z down to the floor. Web search using "offset drain" to see examples. The angles can be sharper than the offsets you will see in web searches showing products called offset drain. You can build your own offset with fittings, and with sharper angles. Having said this, I'm assuming you will have consulted some local authorities for their blessing. The idea is to break the fall, so that water rushing down the pipe doesn't scoot through the P trap entirely. This is because a P trap must have some water remaining in it, which is what they call "trapped" and that make the P trap function as it should. Otherwise it's not working as intended, and it will let sewer gases into your house, and this is a major health risk, a matter of life and death. I'm not a plumber so this entire paragraph may get knocked out later by Master plumbers who may tell you that it's just not going to meet their blessing to plan to install an offset to break the fall. In which case a localized concrete slab break and repair job will be required, to remove that existing P trap. Not a big deal. Easy as pie. You can even learn how to do this by reading on the internet.
2/
Here is one source for trench drains: myshowergrateshop.com
They are less than one inch in height.
http://www.myshowergrateshop.com
The wetroom concept or bathroom floor as shower pan concept is a great idea.
A channel drain (a long line) is great for making a flat (sloped not level) floor inside the shower pan.
Very elegant.
You will still need a slope, so there will still be a threshold to step up over; in other words it won't be a curbless shower -- unless you can figure out how to raise the floor in the corridor outside the shower, or you just cut a groove in the slab for the trench drain.
The shower drain's "male exit" slides inside your existing 2" pipe.
It doesn't matter that this reduces the diameter of the pipe at that one point.
It's not significant, and it's not a code problem.
You might need a bushing since the shower might have a 1.5" exit.
3/
About the extra venting I have no clear answer for you.
4/
The alternative plan you asked about is a no-no, as far as I know. (draining the washer suds into the same pipe as the sinks drain into.)
Whether or not the entire plan is sound from a DWV point of view, is not something I can comment on.
Everything i have written above is partial information.
There is more to know.
In addition to all that i have not yet written, there is a lot of other information that i don't know and which will be important to your specific case.
There are inevitable corrections to what i have written above.
Take no action based on having learned anything in this post.
Get other information from other sources.
Consult local authorities too.
Hope this helps.