Can the order of filtering and downsampling be exchanged? Say for example, I have a signal with maximum frequency at 600Hz, and I originally oversampled it at 2000Hz. Afterwards I was only interested in frequency less than 500Hz. I guess I could first filter out frequency >500Hz, then downsample by taking every second sample point; or I could first downsample by taking every second sample, then filter out frequency >500Hz. Is there any problem with the two methods? Do they generate different results?
In Short: The order does matter.

Detailed:
Downsampling by a factor of 2 without filtering, will cause aliasing in frequencies of 500Hz and more.
Since your signal is bandlimited to 600Hz, the frequency range of 400Hz-500Hz will be corrupted.
So (theoretically) the frequency range 0Hz-400Hz will be the same with or without filtering.
If you were only interested in this range, it would have been possible to avoid the filtering.