I wish you could hear the pronunciation on these questions. Or at least after to answer it.
If you're on mobile, and you click on the comments button, it will give you a listen button in the forum
It rhymes! :)
I have heard native people use "σε" or maybe "σας" before παρακαλώ & ευχαριστώ.
What do they mean for the words?
You are literally saying "I ask" and "I thank", so to complete the thought, "I ask you" and "I thank you".