That's Communism, not socialism. Sure, socialism has some similarities to Capitalism, that's because Socialism was never the end goal, but a transition period between Capitalism and Communism.
Yes, never said that was socialism, only that's how a fair society can only operate. Also, depending on the socialism in the context, it might not even be a transitionary period but rather bourgeois socialism that Marx criticized thoroughly in Critique of Gotha Programme, which is where 99% of my comment comes from
Workers don't 'exploit' themselves, workers decide if it's worth doing the work for what they can reasonably expect in return or not. If they decide it is worth it, it's something they chose for themselves, not even close to the same thing as a Capitalist squeezing every last cent of 'profit' from you as they possibly can and basically forcing you to do it by making it impossible to survive if you don't.
The market forces workers to exploit themselves, given how the primary way cost cutting happens is through decrease of wages, as most production costs are set in stone (like materials needed or machinery upkeep
They still get the full value of their labor. Taxes used to benefit the community benefit workers as well, because they are members of the commuity. Capitalists under capitalism still (are supposed to) pay taxes- and so do workers. Workers who are more fairly compensated for their work and have to pay taxes are better off than a wage slave who isn't fairly compensated and still has to pay taxes.
Yes, but if they pay a portion of value of their labor, they don't get the full value. It's an involuntary payment that is good and necessary, but the point is that getting full value is impossible under bourgeois state and commodity production society