We will be arriving in Mumbai on the 23rd of Decemeber and leaving Kerala on the 11th of January.
Nineteen days isn't a lot of time, particularly when you have some specific activities you want to pursue, and which are available in specific places. I think the decision you need to make soonest is whether or not you include time in Goa on this trip. Three or four days in Mumbai, followed by two weeks in Kerala would work; but equally you could spend some time in Goa before heading for Kerala.
Given your limited time I would recommend travel efficiency wherever possible, and that means overnight trains. I couldn't recommend hitch hiking or self-driving in India.
what I should have said is it would be nice to do some stuff away from the crowds of tourists (if possible).
You are travelling in a peak period to a prime destination; so you might temper your expectations in this regard.
My girlfriend has done a lot of Ashtanga yoga ... We were hoping to find some where to do about 3 days of yoga.
As noted by another, yoga is available almost anywhere. The key question is whether you simply want to maintain a "daily" yoga practise or whether you really want a more immersive experience. If the former, then your hotel or guest house will be able to point you in the right direction (sometimes the roof). If the latter, then some yoga centres, especially at that time of year, will have a focus on longer courses and retreats; you'll still be able to take casual classes everyday however, and might also be able to participate in the non-physical aspects of yoga.
Just as an example: http://www.yogaindiameditation.com/yoga-walk-in-course-at-panmana-ashram-kerala-india.html
http://www.hinterlandvillage.com/
Paragliding, rock climbing and white water rafting sounds awesome.
White water rafting is possible near Vythiri in Wayanad District (Kerala); and also outside of Kannur (Kerala); and outside of Kochi (Kerala) - you might want to check these are not dependent on monsoon rains. Parasailing in Goa (?Baga beach); Paragliding is more likely available in Kerala - Varkala perhaps, and Vagamon near Kottayam. Rock-climing outside of Munnar (Kerala) - and I think in a wildlife park from Panjim in Goa.
