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Yes, they are actually equal... well, sometimes.

Dominance is irrelevant here. "Dominant" doesn't mean "more likely to be inherited." A dominant allele is one that gives you its distinctive phenotype when you inherit just one copy. A recessive allele requires you to inherit two copies before you have that phenotype. That's all. [1] The Punnet square works the same whether the alleles are dominant, recessive, incompletely dominant, or whatever.

Each axis of the Punnet square represents the two homologous chromosomes of one parent's diploid cells. When that cell undergoes meiosis, the homologous chromosomes are separated. One goes into one child cell, and the other goes into another. So each of those chromosomes is present in exactly 50% of the gametes. The same is true of the chromosomes in the other parent. So if you're heterozygous at an locus that codes for something like purple vs white flowers or something like that, half your gametes will have each allele. And if your partner is too, half their gametes will have each gene.

For the purple/white flower gene, that also means exactly 25% of your offspring will have each combination. One gamete at random from each partner. It doesn't matter whether the purple flower gene is dominant or incompletely dominant or whatever.

If the difference between the alleles is something more serious -- for example, not purple vs white flowers, but whether or not you successfully gastrulate -- the allele frequencies in the offspring are going to diverge from 50/50. This is evolution through natural selection. If the "can't gastrulate" allele is recessive, you won't see 25% of descendants in each quarter of the Punnet square. In fact, 0% of the offspring will be homozygous for the "can't gastrulate" allele, because they all died as embryos. However, if we observed the embryos at the moment of fertilization, there would still be 25% of each.

Sometimes natural selection happens before fertilization too. If an allele makes your gametes not work so well, those gametes will be less likely to even make it to the point of generating an embryo. For genes like this, the 50/50 model of the Punnet square kind of stops working. When you think about it, this kind of allele might actually be pretty common. Gametes don't just do gamete-specific stuff like making flagellae. They also need to do normal things like transporting glucose into the cell, and if they are worse at any of these normal things, they're probably less likely to succeed at being gametes. On the other hand, they probably don't even express the gene for flower pigments, so that one is probably truly 50/50.

[1] In reality, a lot of alleles aren't really 100% dominant or recessive. If you are heterozygous, you may get a phenotype that's somewhere in between the homozygous phenotypes. Sometimes it's exactly halfway between, but it is often very close to one side or the other, and those genes are considered dominant and recessive, even if they aren't truly 100% dominant or recessive. But none of this is relevant to the frequencies in the Punnet square.

They're not biologically weighted; it's just probablilities. Punnett squares are just models for predicting inheritance. Like all models, they are indeed not 100% accurate, and the 25% weight is not true for all traits.

Some traits are more dominant than or can blend with other dominant traits. We see this a lot in flower colours.

For autosomal traits that arise from combinations of genes, you can able to use a Punnett square, but you need a bigger square. For example, a trait that comes from 2 genes on different chromosomes would need a 4x4 square, and the probability of passing on each copy of a gene drops to 1/16 instead of 1/4.

Generally yes, and the other answers already do a good job of explaining this, but I want to point out that there are real-world cases of genes which do not follow this model (and not just due to post-zygotic lethality of some genotypes as already mentioned). This is called segregation distortion or meiotic drive, and involves one allele (or often a group of linked alleles) increasing its own chance of being passed on at the expense of alternate alleles. Mechanistically, there are various ways this can work, including such a gene ensuring that it ends up in an egg cell rather than a polar body during oogenesis, or in some cases actively killing any gametes which don't share the same allele.

Besides these cases where meiotic drive is inherent to part of an organisms' genome, similar inheritance patterns can also be manipulated by parasites or pathogens. Wolbachia bacteria infect insects and are passed on from mother to offspring via eggs, and have famously developed ways to manipulate their hosts to produce only female offspring so that they don't end up in a male (which would effectively be a dead end). Humans have also been experimenting with the use of meiotic drives for artificial selection purposes, including some recent high-profile studies into the use of such a mechanism to reduce the ability of mosquitoes to transmit malaria.

Oh I think part of it is how you think of “dominant” and “recessive” alleles. I feel like I was taught that dominant alleles mask the recessive allele - so in other words as a random example that the allele for brown eyes is dominant and the allele for blue eyes is recessive, so if you have just one copy of the dominant allele it will hide the blue eyes trait. Often that’s not what happening - really the dominant allele makes a functioning protein while the recessive allele makes a protein that is “broken “ and doesn’t do anything. So the recessive phenotype is really the result of not having the functioning protein at all. Obviously there are tons of different scenarios, but a lot of genes do work this way.

I have no idea if there is really a gene for blue and brown eyes but using that as a hypothetical example: the dominant allele that gives you brown eyes would really be making a protein that does something with pigment that turns your eye color brown. Without that protein doing it’s job your eyes would just be blue. Having one copy of the allele that makes the functioning protein will make enough of it to turn your eyes brown, so it’s “dominant”.

I think your question comes from a place of both a misunderstanding of what dominant and recessive alleles are is and what a punnet square represent. Sexual reproduction involves the combining of 2 haploid genomes to form a diploid genome. A haploid genomes contains a complete set of genes, so a diploid has 2 complete sets of genes. All genese have variants known as alleles. If an individual has 2 copies of the same allele, they are homozygous, if they have 2, they are heterozygous.

Dominant and recessive relationships are not about how a gene is inherited, they describe how the phenotype relates to genotype. An allele is dominant to another allele if it masks the phenotype of the recessive allele. For example, in pea plants round peas are dominant to wrinkled peas. This means that an individual that has the round pea and wrinkled pea gene allele (Ie a heterozygote) will have the round pea phenotype. It DOES NOT mean that the round pea allele is preferentially inherited. It is also important to remember that dominant recessive alleles does not describe the majority of traits. It is the first, and simplest, genetic relationship that was observed. Most phenotypes are polygenic, meaning there is a small contribution to the phenotype from many genes.

Next, A punnet square just tells you the expected genotype ratio of homozygous and heterozygous offspring given the parents' genotypes. If two parents are heterozygous for a gene (eg BbxBb) have offspring, the expected genotype ratio for their offspring would be 25% BB, 50% BB, and 25% bb.