1. Compact cameras are just okish when it comes to low light performance.
2. Action, sports photography demands OVF, which almost no compact camera equipped with.
DSLR is the solution for both.
You have a decent budget for a dslr, but you won't get that much zoom in this budget. If you think you can stretch a little, you can get an entry level dslr and the amount of zoom you wanted. But DSLR is bulkier, heavier than the compacts. And lot more you have to consider if you are buying a dslr.
Do you want a DSLR? If not,
IXUS 255 or WX200, makes decent low light photographs. You can handle sports photography using burst mode and have the zoom you expected. Finally, this would cost around 10k.
TZ30, this will do pretty much everything what the above two could + it does have more zoom. Cost about 15k
S110, this will give slightly better image quality and better low light performance than the above three. Cost about 17k
If you really want to hit the roof of your budget, SX50, a bridge camera with plenty of zoom.