Hamlet, I think, is pretending to read from his book, but - as with many of his other mocking comments to Polonius in this scene - he is making a reference to Ophelia.

"Carrion" can mean either dead flesh, or living flesh. What Hamlet feigns to read is the information that the sun can cause maggots to appear in flesh left outside to rot (It's an odd Elizabethan idea that the sun bred all life - now disproved!).

What the coded meaning is is that the son (that same pun Hamlet made in his first scene) might also breed (i.e. have sex with) another sort of "kissing carrion" - a sort of flesh available for sexual pleasure. As the sun breeds maggots in flesh, so might the son breed something else in another kind of flesh... and then he asks "Have you a daughter?".

Though ms-mcgregor is right to assert that, in the early Elizabethan days, many touring companies would tour, it's important to point out that, when Hamlet is written (around 1599) that was less true. Shakespeare actually gives us the reason for the players travelling later in Act 2, Scene 2.

Hamlet asks "How chances they travel?", and Rosencrantz tells him that they don't attract as big audiences as they used to, as a company of "little eyases" (young child actors) are in fashion on the stage. So the players have been forced to tour to make money.

Two excellent questions there! Hope it helps!

Hamlet's purpose in talking to Polonius is to mock the old man but Polonius really doesn't get it. In Act III, Scene 2, Hamlet first calls Polonius a "fishmonger" or a pimp. Hamlet is really saying that Polonius is trying to "sell" his daughter, Ophelia, to him. Then he makes his comment about the sun and maggots followed by "Have you a daughter". When Polonius says yet, Hamlet adds,"Let her not walk i' th' sun. Conception is a blessing,/but not as your daughter may conceive. Friend, look to't. Many critics think this is a play on words between the word "sun" and "son". They believe Hamlet is referring to himself as the sun/son and making a rather vulgar insinuation about Ophelia. However, Polonius never understands the pun.
To answer the other part of your question, the company of players is doing what most actors did during Elizabethan times. There were very few standing theaters ( the Old Globe was London's first) and so actors had to travel from place to place in order to find places to earn money by performing.