where can I find a full list of examples of data types in biology?
I am also a little confused about what exactly do you mean by "data type". As far as I know in programming "data type" is something you use to obstruct 0s and 1s e.g string, int and array is the way you'd reference your own data and something that language interpreter/compiler can understand and allocate right memory for. Whereas type of data you can get in biology (bioinformatics), I think is what you are asking. The type of data you get is flat text file i.e ASCII or binary and there are few different binary formats in bioinformatics.

In genomics, which is DNA/RNA type of data you mainly dealing with ASCII files. These are the files that I know of:


FASTQ hold raw - sequenced data. The main two types of sequencing you can do is DNA sequencing (DNA-seq) and RNA sequencing (RNA-seq). Usually FASTQ files always gzip ed and file extension is fq.gz or fastq.gz I wrote this tutorial explaining FASTQ
FASTA can hold different type of sequence data, which is different to raw-sequence held in FASTQ. You can actually convert FASTQ to FASTA, but you'd loose quality information. The two broad classes of sequences you can have in FASTA file are nucleotides sequence i.e DNA/RNA sequence or  amino acid sequence i.e protein/peptides sequence. FASTA is also almost always gziped, file extension varies a bit, most common once are fa.gz or fasta.gz, but you will see fna.gz or fna or faa which tries to indicate FASTA with nucleotides or amino acids. FASTA usually used for reference files e.g reference genome. Here is a link to Ensembl repository for mouse genome
GFF and GTF files these are annotation files, used together with reference genome FASTA files. Here is another one of mine tutorials
SAM and BAM okay here is an example of binary type of data. Sequence Alignment Map, file extension .sam depending on the organism, can be very big. For mouse RNAseq data for example SAM file can be between 1 Gb - 40 Gb, makes sense to compress it. BAM is binary SAM. you need samtools to work with BAM files. Here is pure python BAM parser if you are interested. 


This should cover first two of your points. Although proteomics has few more type of data, again mainly text. Mass Spectroscopy is the tool proteomics people use to get they "raw-data" and there is some finny file format(s) that I'm not aware of and there will be a few downstream file formats I'm sure. 
As for gene expression, that's mainly tabular data, csv or tsv again flat text file. 
One other format that I should mention that covers your point 7 is VCF which is variant calling format here is specifications for it again flat text format.

One thing that I should say; it is somewhat common and preferred, at least in the past to be able to parse biological files with Unix tools such as grep, awk and cut and therefore most of the files are tab separated text files.. I think now bioinformatician coming around and using more structured and compressed data formats. 

This isn't exhaustive list of file format by any means, but I think this should give you enough to google for. 

One last link to Kegg pathway which is your biochemical pathways, they might explain there the type of data they use