Please advise, how to understand the word slave/servant in Judaism? What is meant to be a "servant" of God? What is the origin of this word in Hebrew and its meaning? Has the meaning been changed within time period of centuries, I mean intrepretations or connotations, etc.?

Rav Kook stresses that today, the picture of an unhappy slave is the furthest thing our Loving Father wants from us, and it's even counter-productive, causing more harm than good. That's not part of the modern world anymore, for mankind has, thank God, matured out of slavery. The Hebrew verb EVD means to serve, and just as a popular Prime Minister's aides see it as an honor to serve him, l'havdil, we are honored to be able to serve and be close to God. It must also be stressed that regarding a God Who has no body, "close" isn't measured in proximity, but rather by being "similar" (like a copy is "close" to the original, and we were created in the image of God, "imitateo dei", to copy His traits). As is often the case in Israel, those aides who are close to and serve the Prime Minister, often gradually rise to be ministers themselves, having learned from their mentor. Again, we refer to service out of awe, which leads to striving to realize our Godly potential to emulate Him.