Many thanks for the info my friend!
My pleasure Pal.
Regarding the seacond part of your question... why these two ships are marked with the "13". It seems tha the reason for that is the time these pic were taken and the way of marking of IJN ships at that time. The image of the IJN Hayate was taken during her trials. Because she was launched as the no.13 ship therefore the number 13 is there. And this is the very early way of IJN marking of ships. It seems that the reason for the "13" of the IJN Wakatake is different. I think you copied the image from a site where the pic caption says it was taken in 1920, right? It is a mistake there. How the date could be possible if the ship was launched in 1922?
IMHO, the kind of markings of IJN ships appared in 30'. By that time the markings consisted of two numbers painted on broadsides. The first one usually bigger, was applied at the prow while the second smaller one was painte at the centre of broadside. The first number indicated the Destroyer Division while the second one was the individual number of a ship that often was the number gotten while launching. The system of numbering seems to be a llittle bit changed in 30'. The smaller number was replaced with a kind of writing.
Here you are the IJN Kamikaze destroyer ( the first ship of the class launched in 1922). Please note the number "1" at the prow. The ship was the no.1 of the series.
And here are the IJN Asakaze in 1924 and in 30' and the IJN Hatakaze also in 1924 and 30'. Please note the number 5 of the Destroyer Division for both ships used in 30' but writting looks different. Perhaps the writting could be the ship name. But not sure. The IJN Asakaze was the ship no.3 of the series and the IJN Hatakaze was no. 9.
IJN Asakaze and IJN Hatakaze in 1924
IJN Asakaze and IJN Hatakaze in 30'
And here the group of the Kamikaze destroyers, circa 1926. Please note the two numbers. The right one has the second marking as "15" what suggests the IJN Asanagi. The same , first number is applied for all and indicates the Destroyer Division 29.