Does natural selection occur with intent, or with a pre-planned outcome? It can't, as it is a purely a natural process, just like the diffusion of gas molecules. So then, doesn't it also pass a random by those criteria?
Your 'criterion' is that anything that is natural is random. Actually, this isn't a criterion, you are using this as a definition of "random". When an apple falls from a tree it falls directly towards the center of the earth until it hits the ground. This is a natural process but is not random. The orbits of the planets and their moons can be predicted thousands of years into the future with an extreme precision that is only limited by our patience to make observations and calculations. This natural process is not random. Your definition of random is your own and is not used in any field of mathematics or science. You can define your way into fantasy, but not into reality.
How can the fact that various hands and card draws are not equally probable be irrelevant, but the fact that various phenotypes/genotypes are not equally probable to reproduce is relevant?
I, of course, did not say these things are irrelevant, I said they are irrelevant to a formal definition of "random". I am merely saying that in random processes, the various possible outcomes can have the same or different probabilities. Note that there are only two ways to determine or estimate the probabilities for the various outcomes: 1) to have a quantitative
model of the process from which you can analyze the probabilities - and you are conjecturing or theorizing that this model is a good representation of reality; or 2) to observe the process a large number of times and catalog the distribution of outcomes of the trials.
Also note that a great many processes that we think of and treat as random are really deterministic. When you flip a coin, if you can measure the exact position of the coin on your finger, the exact dynamics of your thumb striking the coin, the exact interactions of all the air currents that interact with the coin, the exact mechanical properties of the floor where the coin lands and the geometry of that landing, etc., you can in principal determine exactly how that coin will land. Most point mutations in genes are caused by the thermal motions of the various molecules in the vicinity of the gene when it is being transcribed. If you knew all these motions exactly, you could again exactly predict the outcome of these mutations. (For the sake of this argument, I am ignoring quantum fluctuations which are, apparently, fundamentally stochastic in nature. But even including such quantum fluctuations, their influence on such macro objects as coins and cards would be small enough that the outcomes of coin flipping and card shuffling could in principal be predicted with almost total certainty.)
The most precise way to describe these processes is that their behavior is such that they are
well modeled as random processes. Whether such a description is appropriate is determined by observation - whether that behavior does in deed emulate the behavior of the stochastic model conjectured. In particular, even when considered as fundamentally deterministic processes, they do not act to produce some pre-planned outcome.