For the plane I think what you suggest will be fine, but be aware that in Japan people, women especially, tend not to dress that casually on the whole and a tracksuit would be seen as very underdressed almost everywhere. Arriving at the hotel from the airport dressed like that won’t raise any eyebrows though I shouldn’t think.
On the gluten thing your first challenge is going to be soy sauce, which can be in almost anything and contains gluten. Gluten free soy sauce is available, but won’t be used as a rule. So you’ll be avoiding almost all readily available Japanese food. Plan ahead and making bookings at special restaurants will be fine in the cities you’re visiting, though language might be an issue, but I think it’ll be really tough away from the cities.
Gluten free in katakana is グルテンフリー, and is easily pronounced guru ten fur ee. And I think wheat is ko mugi. But my hiragana is poor, and kanji worse, so I wouldn’t recognise it on a menu…. However, taberu is eat and so “can’t eat” will be taberaremasen. “Komugi wa taberaremasen” will be I can’t eat wheat. Whether the waiting staff hear you or understand you is another matter, and whether they really understand what that means also questionable, so I’d be wary of relying on asking staff in a regular restaurant to accommodate a gluten free request.
Research in advance and make reservations would be my only advice, though I guess fresh sashimi and plain rice (not sushi rice, which will have vinegar that is likely to contain wheat) would do if starving, or a yakiniku restaurant where you cook the food yourself so can check. Rice crackers from a 7-11 as well?