Is advising friends to pray Tarawih and feeling joy when they respond considered showing off, knowing that the intention is to benefit them with lessons and hadiths I have learned?

If your joy stems from your eagerness to advise and guide your colleagues, and not from showing off or seeking a good reputation, then there is no harm upon you. Indeed, a believer rejoices in fulfilling what Allah has obligated upon him regarding advice, and people are guided through him, as Allah Almighty says: "Say, 'In the bounty of Allah and in His mercy - in that let them rejoice; it is better than what they accumulate.'" And the Prophet, peace and blessings be upon him, used to rejoice at people's response to his command. So continue your دعوتك to your companions and strive for sincerity, for indeed, that Allah guides a single man through you is better for you than the world and everything in it.