Is this a good entropy collector and whitening technique?
The entropy of a PRNG depends on the quality and amount of "true" random noise used to initialise the PRNG and how much bits the PRNG produces before being re-initialised. A fresh PRNG might be as high quality as your entropy collection above, albeit often a different source such as CPU heat.

The method you cite will not add entropy but will spread the entropy fairly evenly within each 256 bit SHA result (Step 3 can be dropped).

If you want to know the level of actual entropy, just save a lot of results from Step 2 and check with a test suite such as NIST's statistical tests.

The key concern you should focus on is not spreading the entropy (even a non-cryptographic hash like MurmerHash will suffice*) but improving the quality and bitrate of the random noise source.

* If the random source is truly random as it should be, a cryptographic hash isn't needed. If it isn't random, a cryptographic hash reduces inferring the signal in the noise.



EDIT

Regarding Step 1 - If you want stick with keyboard/mouse for random noise input, I suggest you use an approach similar to HotBits timing. 

Only the difference between two or three timings (I forget which is used to cancel clock drift) are used to create a single bit of entropy; a mouse could use pixel travel distance per fixed interval. 

As the timings aren't as random as beta decay, you'll need a NIST test suite or similar to decide on how many bits you need to collect for each SHA-256 hash. Hint: It will be much more than 256 bits.