Just go local... cities are not meant to explore with a iteniary... you will enjoy more when you go local... be a part of culture, and talk to local. . you will eventually find the best place , food... Else google is always there for the Top 10/5 places to visit when in Lucknow/varanasi. just a personal thot, happy travelling.
Day1 Fly to Bagdogra and transfer to Gangtok
Day 2 Explore Gangtok visit tsongo lake and moanstaries
Day 3-5 North Sikkim Tour: Lachen, Gurudongmar Lake, Lachung
Day5 Back to Gangtok
Day6- flyback
Its very trek ...Rishikesh- Rudraprayag- Ukhimath- Chopta then trek to Tungnath, Chandrashila. you can book tent or camp, other wise some budget stay available.
In my view, you shouldn't. Four of my friends went for the trek but the day they reached Uttrakhand, glacier burst happened and we aren't able to contact them till now.
Yes you can easily plan it . best route is delhi-gwalior-guna-indore-dhule-pune and Goa whole journey will take around 3 days (depending upon your riding time)
You can choose lake pichola hotel for you stay in udaipur. Its build on lake pichola and offers good view. It restaurant is famous for lal mass and has stunning view.
Check out Bhandardara, dist. Ahmednagar. It's an amazingly beautiful place. There's mountains all around. Bhandardara is situated in the midst of Sahyadri range. You must visit this place once at least. Check out The Land Of Wanderlust (TLOW) website..... https://thelandofwanderlust.com/
Really you are going to travel in best part of India, If you are travelling solo on Bike. Do stay in home stays, better to taste local delicacies from very soft hearten and nice peoples there. With families you can stay in homestays and local cottages.