If you collect data in 512-sample chunks but are constrained to 16-point FFT, you will have to perform the FFT 32 times for each chunk and average the results (either for each chunk or for the entire recording - your choice).
The sampling rate determines the upper limit of the frequency values you assign to the FFT results, and it doesn't matter whether you are looking at 512 samples or 16 samples at a time. Your top frequency is going to be 1/2 the sample rate = 5 kHz.
The series of frequency results will be (in Hz) ...
5000
2500
1250
625
312.5
...
and so on, depending on how many samples you pass to the FFT.
I'm not going to ask why you're restricted to 16-point FFT!