Hello there!

Am particularly delighted to do this research for you since I am headed to Bhutan in about a month myself and am keen on some of the same things you are looking for as well. :) Bhutan is set in the majestic Himalayas and is steeped in culture and a sense of tradition that sets it apart from its larger neighbors. I am sure you will have a memorable time there.

Weather in Bhutan in February

Temperatures in the far south are 15 degC in winter (December to February). In Paro it is -5 degC in January

The winter is a good time for touring in western Bhutan, bird-watching in the south's subtropical jungles, and whitewater rafting. The days are usually sunny, cool and pleasant, but it's quite cold once the sun sets and you will need to pack warm clothing. From December to February, there is often snow in the higher regions and occasional snow in Thimphu. The road from Thimphu to Bumthang and the east may be closed because of snow for several days at a time. It would be best not to plan to visit these regions at this time.

As it is suggested above, places may be closed especially in the higher altitudes owing to snow. I have kept this in mind for the treks and places I have recommended below.

Travel to Paro, Bhutan

Only Druk Air flies into Bhutan from India and connections can be made via Delhi or Kolkata. However Druk Air does not allow for bookings in Feb since its too early, as in indicative price point the following are your options

Indore - Delhi - Paro - For the itinerary below Druk Air has quoted $1440. For a similar flight schedule from Kollkata the price for 2 passengers are $762. However this seems very inflated to me and some sites talk about sub INR 10,000 for a one way trip to Paro from Del/Kolkata but none of the travel sites are throwing any data presently.

Alternately you could fly into Bagdogra and take a bus into Bhutan at Phuentsholing (192kms) and from there onwards to other places in Bhutan. Phuentsholing is the official entry point in Bhutan and permits are issued here, the trip from Bagdogra to here takes about 5-6 hours.  The Royal Bhutanese Government runs a bus service to Phuentsholing from Kolkata (via Siliguri which is very close to Bagdogra). The price for a bus ride from Siliguri to Phuentsholing is sub INR 100.

Indicative prices from Indore to Bagdogra are below. Indicative prices from Indore to Bagdogra are below.

Treks

The Snoman trek is almost the must do trek in Bhutan, However in the interest of time and considering that there might be too much snow in February in the northern parts, this is possibly something you'd like to pass.

Day Hikes Around Thimpu

This Booklet on Day Hikes around Thimpu along with Maps are a great resource. I have picked out two from this book which I feel should capture the natural beauty of the place as well as the cultural aspects

1. DECHHENPHU VALLEY - This is one of the prettiest valleys in the Thimphu area, with abundant bird life. The long hike to Jomkerim Pass leads through pristine forests and offers great views of the
   surrounding valleys.
2. DOCHULA TO LUNGCHUZEKHA/TRASHIGANG GOENPA - Lungchuzekha goenpa is one of the best destinations around Thimphu: beautiful forest,
   and a spectacular view from the goenpa. The loop to Trashigang is easy and pretty. You
   can even get an earlier start (and better views) if you spend the night in the “busy” hotel
   just above Dochula pass.
    

LONGER TREKS

1. Dagala Thousand Lakes Trek - Duration: 08 days of actual Trekking from Paro
   Complete distance: appx. 50km
   Highest point on Trek in metres/feet: 4720m/15,482feet.
   Difficulty level: Medium Easy
2. Haa Valley–Saga La–Drugyel Dzong
   Grade: easy; 2–3 days;
   distance 14.3 miles/23km;
3. Drukpath Trek Grade: moderate–demanding;
   6 days; distance 30 miles/48km;

I have selected most treks around the Paro/Thimpu area considering that you will either fly into Paro or even from Phuentsholing this is not too much of a distance and you might want to do Paro and Thimpu anyways as the main cities of Bhutan.

Tips

1. A detailed tourist map is here for your orientation of the country
2. Entry at Phuentsholing starts at 0830 hrs. Required documents - Valid Indian ID (3 photocopies - Passport, Voted ID, Driving License) + 3 passport size photograph. This process takes about 45mins odd.
3. Booklet on Day Hikes around Thimpu along with Maps

Hope this is useful, for more detailed info on specific trek routes, do's and don'ts and any other detail please let me know.

Regards
Aditi