The weird per-base content is extremely normal and is typically referred to as the "random" hexamer effect, since things aren't actually amplified in a uniform manner. Most people will have this and it's nothing to worry about. BTW, don't let anyone talk you into trimming that off, it's the legitimate sequence.
Duplicates are expected in RNAseq. Firstly, your samples are probably full of rRNAs, even if you performed ribodepletion. Secondly, any highly expressed gene is going to have apparent PCR duplication due solely to it being highly expressed (there is a ceiling on the coverage you can get before running into this, it's pretty high with paired-end reads but not infinite).
You'll see information about the adapters at the end of the fastQC report, but it's probably good to assume that you need to trim them off (at least unless the people who did the sequencing tell you otherwise).
The output from FastQC is most meaningful for sequencing DNA, so you often get warnings and errors when you use it to look at an RNAseq dataset.
Good for you for checking the quality of your data, but be advised that FastQC is extremely strict and will often flag patterns that are not the result of poor quality. Think of FastQC like a quality screen. If it flags something, it means you should take a closer look, but it doesn't necessarily mean that the item being flagged indicates bad data.