AFAIK, for now, there is no "best practises". If I have time I try to do the same workflow with different softwares and compare results.
If I have only a few sequences (e.g. genes) and not the whole assembled genome, Can I make a differential expression analysis?
I assume you used cDNA capture or related capture technique to extract your RNA of interest. If so, it is totally fine to do gene expression analysis using your data.
RNA-seq libraries: fastq files from RNA-seq experiments with the following conditions: Control, Treatment 1, Treatment 2, Treatment 3.
You only have n=1 ? The statistical power of your experiment will be very low, becareful on results interpretation
First, take a look at your reads quality using fastQC or fastp to have an overall look at your sequencing
You can align your read in a reference genome (complete genome) to check if your reads are falling into your gene coordinates, which will be a good check validation of your capture.
If you are aligning to a genome, do not align your reads with a non slipce aware aligner as Bowtie2 without specific options. With default option Bowtie2 is not aware of splice events you will have in your genes, prefer HISAT2 or STAR. Also, you can take a look at pseudocount software like Kallisto or Salmon. If you are aligning on a transcriptome Bowtie2 will be ok
For the counting part you can use featureCounts or HTseq, or use pseudocount with Kallisto and Salmon.
If you want to look at expression variation between gene A and gene B in the same condition, TPM normalization will be enought.
If you are looking at variation of gene A across conditions, tools like edgeR, DESeq2 or Sleuth will help you.
See also for normalization : RNA-seq, why normalize for library size?
A couple of points; 3 replicates is the bare minimum. DESeq uses information from all the genes to estimate dispersion, that step might be a little strange with only a handful of genes being measured.
With only 5 genes of interest, why aren't you using qPCR?
because I'm working with public data with a lot of different species....
So does that mean you're not actually going to perform the sequencing yourself, but you're going to download data that other people have sequenced and deposited in a public repo?
Your working with lots of public NGS data with only 5 genes?