What makes recessive genes weaker than dominant?

They are not necessarily "weaker."  They are simply a form that creates a different phenotype that is overshadowed by the other allele.  For example, if all members of a certain plant population were homozygous for an allele, r, and produced no pigment in the petals, the flowers might all be white.  However, if r mutates to R, which produces a red pigment, then any plant that has an R allele will have red flowers;  the R overshadows the r.  In terms of natural selection, a mutated allele may be selected for IF selection pressures are applied;  this is one of the bases for evolution.  Otherwise, it may be a case of a neutral mutation, and there is no selection for or against the new phenotype.  Free and attached earlobes in humans may be an example.

Exactly as the Asst Prof says: the recessive gene simply doesn't code for the production of a protein, but the dominant one does.  Therefore, if you have either one or two copies of the dominant you will have the protein present; the only way you don't have the protein present is when both genes are recessive, that is, non-coding for the protein.

For brown eyes, the gene codes for the melanin production.  The only case where the production fails is when both the gene copies are recessive (light eyes).  The finer colors, controlled by other genes, can be overridden by melanin simply because it is a "dominant" color.  If either melanin gene is present, it doesn't make any difference whether the genes for green are present or not; you'll get brown-to-black.

No, each genotype results in a phenotype that is adaptable to the environmental niche in some fashion.  A dominate trait may have an advantage over a recessive one in a particular environment, or vice versa.

What is the characteristics of a recessive gene?  Here's what I think:  There is the pure breeding BB genotype resulting in Brown eyes.  A mutation occurs at the BB allele and alters one of the dominate B genes so that it is dysfunctional, or partly so.  The result might be  Bb allele, but the phenotype is still brown eye color because of the one good B dominate gene.  In this case, the mutated 'b' gene either does not transcribe a usable protein or it doesn't transcribe any protein at all.

If some one carrying the Bb allele genotype crosses with someone else carrying the same mutant form allele for eye color, the dominate gene B will be expressed (on average) over the recessive 'b' in 75% of the offspring, with 25% expressing the default recessive blue-eye trait.

If a pure breeding recessive bb allele crosses with another pure breeding recessive bb for eye color, the resulting phenotype will be blue, and all resulting offspring from the cross will also be pure breeding bb allele for blue eyes.

The point is: a recessive gene is simply a once dominate gene that got mutated such that its protein product doesn't fold properly to fit into (match) its intended target receptor molecule. The target receptor molecule never gets activated as a result.  The recessive gene allele is either too dysfunctional to work, or the recessive gene isn't physically present there in the allele. I hope that kind of makes sense.

Best regards