You're not seeing bins because you're supposed to take your rubbish home with you rather than ditch it in the street
Seriously though I had no trouble finding bins in public spaces such as train stations, shrines, gardens, etc. If you've been carrying your trash around for a couple of hours, drop into a train station and see if you can't spot a bin - or just outside Metro station exists rather than inside the station.
You might also be missing bins for recyclables - rather than a one-hole-fits-all, these have specific compartments for paper, PET, cans, etc.
Basically just keep the first plastic bag you get and stash your trash in it. Sometimes you have to hang on to it until you get back to your hotel, other times you can get shot of it sooner. It's no real trauma.
I'd use "Ja, mata ne" for "see you", but dewa is equally suitable. "See you tomorrow" would be "mata ashita".
There are two counting systems in Japanese between 0-10. From 11 there is only one. If you want to use the ichi, ni, san... system, you should really add a counter type to the number, but it's not essential for a foreign tourist to get that right. I would also use o kudasai rather than onegai shimasu when asking for a physical item as opposed to an ephemeral (a service, for example). Thus:
grande cappuchino o ni-hai kudasai.
(where -hai is the counter for liquids in cups / bowls / glasses, etc. Not bottles, which is -pon or -hon).
In the other counting system - hitotsu, futatsu, etc - the number itself is all you need. No counter is required:
grande cappuchino o futatsu kudasai.
"mata" is the only real informal "goodbye" I know, but that doesn't mean there aren't others. Also, people might be saying shitsureishimasu, which more kind of means "Please forgive my impoliteness", if they're leaving at a time which they consider to be early, or if they perceive that they are cutting you short by leaving.
Depending on how many more days "a few" is, are you there long enough to catch the opening ceremony for the sumo? That might be worth a shot
It opens on the 10th of May:
Ticket Sales Information - Nihon Sumo Kyokai Official Grand Sumo Home Page
I heartily reccommend the Edo-Tokyo Museum:
Edo-Tokyo Museum
If you like cats, head up to Ikebukuro and go to Tokyu Hands' 8th floor:
http://ikebukuro.tokyu-hands.co.jp/
Sengaku-ji is a fantastic way to spend a couple of hours:
Tokyo Travel: Sengakuji Temple