What unit do I get on the y-axis of a metagene profile plot?
Feel free to @ me in deepTools questions, since I'm the primary developer.

For a given bin, the count assigned to it is the number of reads overlapping it, regardless of whether they overlap by 1 or 10 bases. So a read overlapping only partially and one overlapping completely are treated the same.

Since your bigWig file is in units of "alignments" (i.e., it's not 1x normalized), the resulting profile will also be in units of "alignments" (i.e., profiles and heatmaps are in whatever units the input files are in).

Upstream/downstream regions and unscaled regions are also 10 base bins. Note that these are then the average of the per-base value, since the bins here may not perfectly correspond to the bins in the bigWig files. The line in the profile plot is indeed the average (by default, you can choose median, max, min, etc.) of the underlying regions for each bin.

Regarding the scaled section in the middle, the the number of genomic bases per bin is changed such that the region will have "length"/(regionBodyLength/binSize) bases each. As above, the per-base value is then averaged (or whatever you specify) to derive the per-bin value. The length here is decreased if you have unscaled regions, since otherwise bases would get counted twice.