It is a "Japanese" console as it plays Japanese games with no issue. But it's wrong to say that it's a console that is sold to and used by the Japanese.
You'll probably never run into the software region lock at all, as the Sony PSBBN (The predecessor to the PSN) that used to host downloadable content for the PS2 is now mostly defunct and useless ever since the PS2 lost support for it's HDD unit.
Great! :)
Probably not. AFAIK all PS2s after the v7 (Early SCPH-39000 series) were manufactured in China.
Consoles starting from the v8 onwards (E.g. my SCPH-39006) were manufactured in China.
Older consoles were all manufactured in Japan, with the oldest versions of the console (All v0 consoles; The SCPH-10000, SCPH-15000 and SCPH-18000) even bearing their factory names.
All slimline consoles are the same (As in the construction and software on these consoles is universal), with the only difference between the models being the console's drive MECHanics CONtroller(MECHACON) chip, which determines the region and the behaviour of the console.
As I mentioned above, all NTSC-J region consoles will play Japanese games (Including the console you have).
The only thing that will probably not work (And which you will probably never ever encounter) would some Japan-only software that was available on the PSBBN (And the PSBBN client itself).
It was part of a now dysfunctional system to deliver downloadable content to home users.
The button used for selecting options varies from software to software for the PS2. As I mentioned above, the select button is CROSS for that PS2's OSD (The PS2 browser).
If you play a Japanese game, circle will be used to select options and cross will be the cancel button.
Playing a USA/NTSC or Europe/PAL game on the console instead will mean that cross is select and circle is cancel, in-game.
There is no generic fix for this, as the developers of the game are the ones who determine which buttons will be used.

