which one file to choose the adapter sequences to use and what are the value to choose for Maximum mismatch count which will still allow a full match to be performed ? How accurate the match between the two 'adapter ligated' reads must be for PE palindrome read alignment ? How accurate the match between any adapter etc. sequence must be against a read? and in trimmomatic opretion .
How to perform QA on your data depends on the fastq content and the downstream analysis to be performed.
Many of the Galaxy tutorials cover QA steps and can be found here: https://galaxyproject.org/learn/
Trimmomatic is one choice for trimming. The tool itself (the algorithms) and parameters (the usage) are described here and most options are implemented in the Galaxy wrapped version of the tool: http://www.usadellab.org/cms/index.php?page=trimmomatic
Hope this helps! Jen, Galaxy team