Digital filters in LabVIEW
Let me answer your two questions in turn:

For your first question, generally, yes that is correct; if you filter a 10KHz sinewave that has a DC offset with a filter that has a cutoff frequency below the frequency of the sinewave, then the sinewave would be rejected. The amount of rejection specifically depends on the performance of the filter, but given you said you have a 1KHz cutoff frequency, the sinewave is significantly higher and therefore sufficiently rejected. 

I see in your plot that the order of the filter is 5, which for a Butterworth filter as also shown would have a rejection of 20dB/decade *5 (where 5 is the order of your filter), or 100 dB per decade. Depending on other factors such as your digital dynamic range, this suggests that you would be able to filter your 10KHz sine wave up to 100 dB (10KHz is a decade above the cutoff frequency).  The DC signal, which is below the cutoff frequency would pass through to the output, unless something in your system blocked DC or introduced other DC -offsets (which is possible).  Also the filter itself can have gain or loss, so the actual DC output level if it did pass through can be modified by this gain or loss accordingly.

For your second question, sampling frequency is the sampling rate for the signals passing through this digital filter implementation.  From the figure, you are using a sampling rate of 200KHz, and yes this would be the sampling rate of the sinewave that is created. 

I hope this helped to clear up some of your questions.