It can be a little confusing. A genetic allele is just a portion of the genome responsible for an observable trait or phenotype. Examples of phenotypes consist of your example (eye color) and millions of others. As @swbarnes alluded to, most phenotypes are complex phenotypes, meaning they are not described by a single genetic locus. Height, weight, hair color, pulmonary function, and eye color are all examples of complex phenotypes (or traits); meaning that more than 1 genetic locus controls the observed phenotype. Other traits like PTC tasting bitter are determined by a single genetic locus; these are referred to as simple traits.
When it comes to dominant verses recessive, this classification is usually applied to simple traits, and refers to whether it requires two copies or one copy of a causal genetic variant to observe the phenotype. For example, a single copy of the genetic variant responsible for sickle cell disease does not cause the phenotype of sickle cell anemia. It takes two copies to cause it. Thus we label this genetic variant as a recessive allele. If a single copy of the genetic variant were sufficient to cause the observed phenotype, then we would label that variant a dominant allele.
Some traits are intermediate, in that a single copy of the allele causes an intermediate phenotype, that is more exaggerated if you inherit two copies. The PTC tasting bitter above is such a trait. If you have a single variant allele, it will taste bitter, if you have two copies of the variant allele, it will taste very bitter. If you have zero copies, you will wonder why your friend with two copies just made a horrible face.
IF you are referring to a simple trait (controlled by only one genetic region) and the allele controlling the observed trait is a dominant allele, this would mean a single copy of the variant allele is sufficient to see the trait. However, this also means that both parents could be heterozygous for the allele ( each possessing 1 copy of the variant and one copy of the normal). In this case, there is a 25% chance their offspring gets both normal alleles. So your supposition was correct if the trait is simple and dominant.
With regards to 50% chance of passing any given allele, yes. However, in complex traits, the math gets complex quick with regards to odds of any given phenotype.
Lastly, some traits are influenced by both a genetic determinant and an epigenetic determinant that can be environmentally driven which provides infinite more complexity. I'll let look you look that up or ask another question :) The same with "gene drives".
A neat listing of simple traits in humans and whether they are dominant or recessive comes from here: