Shortest Answer: No.
Longer Answer. No way. That software is extremely old and outdated and no longer supported.
Just keep in mind OBIEE 10g is very very very ... very old !
Therefore you can't expect it to work perfectly on new browsers, the certifications covers minimal versions and the "+" generally means up to the existing version when writing/publishing them.
From a client point of view the OS doesn't matter as you are using a browser, if you mean for the Admin tools which are needed, you can forget Windows 10 (even 188.8.131.52 doesn't work on Windows 10), and I would doubt Windows 8 will make it. You are already lucky that Windows 7 seems to be certified.
Actually OBIEE 10 ran like a charm on Windows XP...
1013210 this is probably where your confusion comes from. 10g was a hybrid between Siebel times and Oracle times and before the switch to a full enterprise architecture.
That switch which happened in OBIEE 11g turned the tool into a correctly managed server farm and made it impossible to run on any random desktop operating system that's better suited to Excel and playing Minesweeper.
The only thing these 2 (Windows 10 and OBIEE 10.1.3.4.1) have in common is the number 10.
This version of OBIEE is over 7 years old according to - https://docs.oracle.com/cd/E10415_01/doc/bi.1013/e10417.pdf and a lot has happened since then. As Christian & Gianni have already said, you'll probably need to consider a newer version of OBIEE as this version isn't even supported any longer by Oracle as far as I'm aware.