když ptáčka lapají, pěkně mu zpívají– when they are trying to trap a bird, they sing to him nicely. In other words, they are ready to tell the bird whatever it is that he wants to hear in order to catch him. This means that people often lie to get what they want.
Co oči nevidí, to srdce nebolí - What the eyes don't see, doesn't hurt the heart. What you don't know won't hurt you.
Ranní ptáče, dal doskáče. - An early bird, will get further. The earlier you wake up, the more things you will be able to accomplish.
Darovanému koni na zuby nekoukej. - Don't look at the teeth of the horse that you got as a gift. I assume, that the meaning of that is "appreciate any kind of gift you receive"