Particularly it seems that I have one range of files corresponding to a HiSeq sequencing: For which I have a forward and a reverse file, and for each of them I have a PE and a SE file.

This is a bit fuzzy. Paired-end (PE) data has two files (with R1 and R2 in the name to indicate the files containing reads from two ends of fragments sequenced). I am not sure why there is single-end (SE) data unless the samples were separately sequenced separately.

These files are under a folder named "cleaned", when I also have a folder named raw, containing only two files, a forward and a reverse. I would guess it is the "Pair end" and "single end", but in this case, I don't understand why I have both reverse and forward for the Single end sequencing?

Raw folder should be the untrimmed data. Second part does not make sense (reverse and forward for the Single end sequencing).

I also have the same type of files for what seems to correspond to a MiSeq sequencing.

There is nothing special to indicate the sequencing would be MiSeq unless these are 150+ bp long reads, lengths of up to 300 bp only possible on MiSeq. It is possible to identify data as being from MiSeq by looking at the barcode of the flowcell (which should be in all read headers).

Besides, I also have two folders containing LJD, for Long Jumping Distance files (processed and processed_cleaned) And I have 2 ranges of files in them: some "3kb" and "8kb" and for each of them, a forward, a reverse and a singleton. I am not sure to understand if LJD is another way of HiSeq sequencing providing longer reads or type of assembly. And in any case, I don't understand what is the difference between 3kb and 8kb. Are these different files supposed to be used together or are they different treatment of the same obtained sequences

These could be mate-pair libraries with longer inserts. There may be two insert lengths.

Overall if this all is for a single sample then someone seems to have gone to great lengths to create comprehensive libraries/sequencing which would be useful for doing assemblies.

You should definitely ask these (and other) questions to the people which gave you the data. Fastq files are very poor on metadata, and what you can guess from file and folder names is very limited. Anyway:

1) you have not received a genome, you received sequencing reads from a genome.

2) Cleaning (removing contaminants, adapters, and low quality bases) may leave orphaned paired reads, which are typically placed on a single SE file, but may be placed on two SE files, one corresponding to R1 and other to R2. You have to ask how the files were cleaned.

3) How do you know the reads come from HiSeq and MiSeq? Folder names?

4) 3kb and 8kb are the expected insert sizes for the LJD or mate-pair libraries, useful for scaffolding genomes.