Should the questioner continue inviting non-Muslims via Skype using phrases like "are you tired from this life, come here http://english.islamway.com/ and you will lose no thing" and similar expressions, and are there other callers to Islam who use such programs for Da‘wah?

A Muslim is permitted to invite to Allah the Almighty to the best of their ability, but they must not delve into matters of which they have no knowledge. They should strive to acquire religious knowledge to fortify themselves and increase their faith. In their دعوتهم (call/invitation), they should focus on explaining the merits of Islam and that it is the cause of success in both worlds, and they may utilize messages, books, and websites to convey the invitation.