The current fashion is subgenes (domains): https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3258039/

No, the collection of genes is. There are exactly zero individual genes roaming around replicating in isolation.

Sexual reproduction uncorrelates the selective pressures over each gene, and each small set of them (with larger sets being possible the larger a population the species has).

The gene is the unit of selection is perfectly right. It's missing a finer point in that groups of genes are unities of selection too, but it's correct.

Not according to modern understandings of gene theory. The Selfish Gene by Dawkins is perhaps the seminal work on this topic for the popular audience.

This is a common misconception. The collection is the unit of replication, but the gene is nonetheless the unit of selection. This argument was laid out in excruciating detail by Dawkins, but the TL;DR is that reproductive fitness can only ever be measured relative to some environment, and all the other genes in a replicative unit are (part of) the environment for a gene and its alleles.

Collection, chromosome, Gene, base pair. Selection happens at all these levels.

Nope. Selection can only happen at the level of something that has an effect on the phenotype. That's the definition of a gene.

Actually, it's chemical enviromments.

How does a chemical environment replicate and mutate?