Why does plasma protein saturation of a drug increase its volume of distribution and clearance rate?
First a couple of clarifications:


You wrote:


For the distribution volume to increase, unbound drug concentration in plasma should stay constant while tissue drug concentration has to increase when more drug is administered.

Actually, it is the total drug concentration in plasma (bound plus unbound) which matters while calculating distribution volume.


Clearance is better defined as $ \frac{\text{Rate of elimination}}{\text{Plasma concentration}}$. The equation you mentioned above is a derived formula for calculation of dosing rates.


Now for the answer. The basic principle is: when plasma proteins get saturated, increasing drug concentration increases both tissue concentration and unbound drug concentration in plasma, because the drug has "nowhere else to go". 

Thus your reasoning is correct when you say this:

The reason why volume of distribution increases is because the drug is distributed to tissues other than blood once the plasma proteins are saturated with the drug.

Of course, our understanding here is that the increase in total plasma concentration is marginal as compared to the increase in tissue concentration.

However, the following does not seem to be correct:

As for clearance rate, it increases because the zeroth order reaction is always faster than the first order reaction in Michaelis-Menten kinetics.

(Perhaps you are confusing plasma proteins with drug metabolising enzymes?)

The answer to the clearance part can be obtained when you recall that we are talking of a drug with low hepatic extraction ratio. Thus, clearance is proportional to unbound drug concentration. As already mentioned above, unbound drug concentration increases on increasing drug concentration.

Hope that answers your queries.

Reference:

Buxton ILO. Pharmacokinetics: the dynamics of drug absorption, distribution, metabolism, and elimination. In: Brunton LL, Hilal-Dandan R, Knollmann BC (editors). Goodman &amp; Gilman's the pharmacological basis of therapeutics. 13th ed. New York: McGraw-Hill Professional; 2017. pp 13–29.