Well, learning Dutch might be not too difficult for you. Although there are lots of differences between your mother tongue and mine, there are similarities as well. I'm working at an scientific institute which is equipped with an international staff, people are literally from everywhere over the world. And.. most of them are perfectly able to learn Dutch, but... it takes some effort. So, try your newly lerned Dutch on some native speakers... I like to do that 
(and yes, our English is fine and we like to show that to foreigners... so quite a lot of people will conversate in English to you as well).
I'm not from the area of Den Haag/Wassenaar, so I cannot recommend you a stable or riding school.
Winter is usually the wettest season in the Netherlands and as long as almost the whole western part of our country is below sealevel... countrysides are therefore wet in wintertime. Not every farmer likes it to have horses outside, where they easily trample all grass to mud.. So that's why a lot of horses are inside during wintertime, for whole days or parts of it.