Tokyo Station will be great having wide range of hotels from high end to budget ones.

Be sure to check out Happy Cow to look for vegetarian good.

https://www.happycow.net

Note that just because you see a dish with veggies, it will not necessarily be "vegetarian." Soup stock is made from chicken, beef, pork, fish.... and/or seaweed (which is the only vegetarian choice). Shojin ryori (temple food) is vegetarian. Soba and udon (with the soup), Kaiseki, and most other Japanese cuisine are not. You can find rice balls (umeboshi - pickled plum, and konbu - pickled kelp), nice salads and tofu at convenience stores, Definitely try kimpira gobo -- sesame oil pan fried burdock root. If in a pinch, look for Italian or Indian/Nepalese restaurants.

https://www.neverendingvoyage.com/vegetarian-survival-guide-to-japan/

T’s Tantan within Tokyo Station and Ueno Station offers completely vegan ramen.

We always prefer non-smoking hotel rooms as one of us are very sensitive to prior smoking in the room. While searching for hotel rooms in Japan, I noticed that only occasionally rooms are explicitly marked as "non-smoking". If a room is not marked "non-smoking" should I assume that it has been used as a smoking room in the past? Quite often a room is not marked smoking or non-smoking. In such a situation, can I go ahead reserve the room and then request a non-smoking room, or is it not advisable?

Is Minato-ku acceptable area for tourist based on above consideration specifically public transportation?

Depends on where in Minato-ku. It covered a large area with many districts in it.

Sheraton Miyako Hotel at: 1-1-50 Shirokanedai, Minato-ku, Tokyo, Tokyo-to, 108-8640.

Also, Bunkyo-ku (a different hotel) should also be a good area. It seems fairly close to Shinjuku-ku.

Here's an explanation of the "ku." https://www.fun-japan.jp/en/articles/12806

As for the non-smoking room, it's better to go directly to the hotel website, rather than using booking sites. You can often find better deals and can put in special requests.

There isn't much around Sheraton Miyako. They have shuttle buses going to Meguro station. Shirogane-dai subway station on the Nanboku line is 5 minutes away. Are you using points? Courtyard Ginza is much more convenient.

Depends on where in Bunkyo-ku. I wouldn't consider it close to the heart of Shinjuku.