Most genomic libraries don't have problems with adaptors. They only crop up when the sequences that one wants sequenced are very short. You probably have 60-100 bases of a 200+ base DNA insert, so you won't see adaptors.

Usually, duplicates are figured out after alignment, not before. Computationally, it's easier on a sorted .bam than on raw reads, if you go by coordinates.

The fastx toolkit provides some simple to use command line utilities to do this.

Check Tagdust from: http://genome.gsc.riken.jp/osc/english/dataresource/

If you need to do the oposite (select fastq reads with certain pattern) there is fqgrep: https://github.com/indraniel/fqgrep

Filtering for unique sequence is a very bad habit I never understood why people would even envisage doing that: you enrich for sequencing errors.

You can use FastQC to figure out if you have adapter issues and also base bias.