In U.S. usage: "I put a lid on the cup." "I took off the lid" or "I took the lid off" or "I removed the lid." "I put the lid on again." For the last case, colloquially, I would probably say "I didn't get the lid on right, and it leaked." Or "I didn't get the lid on completely, and it spilled." In the cases, that's sloppy, incorrect usage because grammatically, the pronoun "it" seems to be referring to the lid, but nevertheless that's what I'd say. "Leaked" or "dribbled" suggests just a few drops, not a big problem.. "Spilled" suggests the tea spilled on the floor or the table, making a big wet spot that ought to be wiped up. If I spilled the tea on myself, I would add that to the description: "I didn't the lid on well, and it spilled all over me."

1. It's called a lid :) 2. Took off the lid, yes. 3. The 'on the' twice is unnecessary. "I put the lid on the cup" or "I covered the cup" is fine. 4. Better to say you didn't put the lid on properly.