I thought that in single end we sequence only from one end, say P7, whereas P5 is attached to the flow cell. Also, after amplification the reverse reads are washed away, meaning we have only forward reads. So, there is always only one primer for each sequence.
I have forward and reverse single end reads. I can distinguish them from fastq file according to a forward or a reversed primer. However, I do not understand how one can get reversed and forward single end reads from the sequencing. If I look at the forward reads in FastQC, the per-base quality looks good, if I look at reverse reads, the per-base quality degrades faster.
I thought that from the sequencing we get only one read of the form: forward primer - fragment of interest - reverse primer as exactly on this pic below. Why in fact do I have reverse and forward reads identified by primer then?

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I do not understand how one can get reversed and forward single end reads from the sequencing
Probably I don't fully understand your question. When you ligate the Illumina adapter P5 and P7 the insert may be in one orientation or the other each with 50% chance. This is the reason why in a "standard" DNA-seq library you get ~50% reads aligned in forward and ~50% in reverse. (Right?)
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Your description suggests that you actually have paired-end reads (otherwise, it's hard to envision how you would obtain separate FastQC metrics for forward and reverse reads). A few questions to clear up the confusion:
1) Do you have one read file or two? 2) If two, what are the names of the files? 3) Can you post the first ~12 lines of the file(s)?
Paired-end sequencing determines the sequence of each end of the target sequentially, with resynthesis of the template between reads one and two. The Illumina website offers detailed descriptions of the technology.
Also, the reads begin with the target sequence, using the forward and reverse primers as initiators, not at the extreme ends of the library as you have diagrammed.