It's not sport but bouldering - check out Hampi, India. Tons of boulders, cheap relaxed atmosphere.
I've trekked in Langtang and Annapurna, Nepal. With the addition of Everest basecamp those are the only regions with teahouses for the "classic" Nepal trekking experience. Everywhere else will likely require a tent at some point and probably a stove and carrying food. For the teahouse treks you can get by with a down jacket, shell jacket + pants, sleeping bag and cash. I cary these everywhere I go in the world. If you don't bring them all of those you can buy in Nepal. Good quality stuff will cost the same at does at home. The cheap shit is cheap; rental boots are blister mongers. Most of the teahouse treks could be done without a sleeping bag - but everyone I met who did this ended up with tons of bedbug bits.
When are you planning to do this? If it's not good, clear, weather trekking in Nepal is less than optimum; half of why you are there is good clean views of 8000m peaks May-October doesn't provide this. During that time there are awesome things to do in India, Pakistan, China or Central Asia (kyrgyzstan/kazakhstan) those are the places that you are more likely to find what you are looking for