Firstly, from what I understand, it is uncouth to perform DE analysis on TPM values - I still don't understand why, since the values are normalized for library depth and transcript length, aren't these the best values to perform DE analysis on?

TPM is a normalization, indeed, but it's a very simple one. DESeq2 is smarter, will do a better normalization, and therefore needs the raw data for properly estimating the dispersion. If you already "pre-normalize" a part of the information is lost (e.g. library size).

Secondly, is there a tutorial for importing Salmon files using tximport package into R that's written in layman English. I tried the Galaxy-based DEseq2 and picked the TPM option, but was informed by one of the moderators that the online tool has never worked properly with TPM values. So, I am left at the mercy of R, which I am not too friendly with.

I find this tutorial pretty easy to follow, and I really don't like R: Importing transcript abundance datasets with tximport.
This bioconductor workflow RNA-seq workflow: gene-level exploratory analysis and differential expression is also a good starting point for DESeq2.