yes. mirrorless cameras are perfectly fine at doing all of that and more.
a6000 is one of the best cameras out there under 1k. the 11fps with 150+ focus points is AMAZING at action shots. it's about 650 in canada.
why do you think mirrorless cameras are bad?
for non specialized usages, such as shooting in a cave, they're perfectly fine. when you buy a crop sensor dslr, you're pretty much saying, i want the exact same photoquality (if not worse) and i also want it to be 50% to 100% heavier.
I'm with tbc on this one. Mirrorless have become absolutely phenominal. I have the Sony NEX 6 with the built-in 2.4 MP EVF (the predecessor to the A6000).
My good photography friend had a Canon T3i, then 60D, then 7D, then 5D mkii. He never really managed better photos overall because most of it comes down to the photographer (composition, settings, etc) and post-production. He also never really took his kit anywhere due to its massive size.
Oddly enough, Sony makes most of the high-end sensors used in Nikon cameras.
There are really only three reasons people buy a full size DSLR anymore:
1. They have a collection of lenses already and don't want to use adapters
2. They are professionals who stick to the 5D
3. They simply don't know the option exists. To many, the traditional DSLR is the only "real" camera
Have fun shopping! There's tons of good stuff. Keep a little money set aside for some kit, such as a good tripod, and of course, lenses.
Edit: I forgot to mention that my friend ended up selling off his Canon stuff entirely and getting a Sony mirrorless for a variety of reasons.