Why do I always write í instead of ý?
Don't worry, -i vs. -y will keep you occupied for a loooong time. It takes years for school kids to get it all right.
Thanks for the hint. Stupid question: Will learning Czech get me closer to learning Slovak?
It will help, basics are very close, but slovak has some grammar things totally different.
Save it for the plural. :)
With an "í", the "d" sound would become a "ď " sound, which is a different word - as widle said, it's the plural form. :)
Does man and men different in czech? Is there any word for men in czech?
Absolutely, singular words have their plural counterparts and you will meet them soon in future lessons. The plural of muž is muži.