While I am still very new to this, I can give you bits and pieces that I have picked up around the forums. Don't treat it as gospel, as it seems everyone has different opinions and tastes when it comes to mixing juice.
First, if you are shooting for 20mg a day, you might be looking at it wrong. From what I have read, we only absorb about 10% of the nicotine we inhale from cigarettes and vapor. So if you are planning on vaping 20mg worth of nicotine liquid, you may only be getting about 2mg of nicotine. I think the popular consensus is to start on a low mg juice and increase until you find that place where you are comfortable with. Some might use a 11mg juice and vape all day, while others might use 36mg and just vape now and then to ease their cravings. It is really different for each person. For myself, I use a 36mg strength and go through about 2-3ml a day, depending on how much idle time I have. I was a 1 1/2 pack a day smoker. I find that this amount completely fills my need for nicotine.
As far as cutting, if you did 1 part of TW to 3 parts of PG/VG I am positive you would see a large decrease in flavor. That can be resolved by adding your own flavor. There are lots of tips and tricks in the forums for different mixtures, you will just have to find what works for you. You might want to do a search here for "tobacco absolute", as it will probably do what you are looking for. It is a flavoring that, when well diluted, has been well received by the community. You might try something like this:
10 parts 36mg juice of your choice
25 parts PG/VG/Alcohol mix (I use 60%PG, 35% VG/5% Everclear) or just PG or VG, your choice
3 parts flavoring (menthol mixture, Loranns, etc.)
2 parts tobacco absolute diluted mixture
It will take some experimentation on your part to find that "perfect" juice to meet your taste needs, but that is half of the fun
Adding menthol flavoring was a question I asked and got some good info on. In short, you can add menthol to any mix and get the results you are looking for, you just have to find the right quantity for your tastes.
Hope that helps! I am sure many more people will be able to chime in with their own experiences.