Well, it's certainly true that Gates and Allen developed Altair BASIC on a PDP-10 (under which operating system I don't know, but it could have been TOPS-10, and you are right that TOPS-10 is not a version of TENEX), but they didn't write MS-DOS; Tim Patterson, at Seattle Computer Products, did. And it's a straight copy of CP/M, not of TOPS-10. (Nor of TENEX.) Later development of MS-DOS happened at Microsoft, but I don't think Gates and Allen wrote any code for it; the last product Gates wrote code for was the BASIC interpreter for the TRS-80 Model 100, which shipped in 1983.
I don't think it's accurate to say, "the PDP-8 and -11 OSes would have been inspired by TOPS-10," except in the sense that they were also inspired by the IBM 360 and other systems — ideas did cross-pollinate, of course, but circumstances conspired to prevent especially strong cross-pollination in this case. The PDP-8 predates TOPS-10 by two years, so there is a substantial amount of software for the PDP-8 that could not have been inspired by TOPS-10 — including, I think, early versions of OS/8 — and RT-11 and RSX-11 followed OS/8 rather closely, especially at first. And, as I understand it, inside DEC, the small-systems folks (the PDP-7, -8, and -11) were organizationally separate from the large-systems folks (the -6 and -10), I think in geographically separate locations.
The PDP-6 Monitor came out in 1964 and TOPS-10 was its lineal descendant (the result of progressively modifying the codebase to support different PDP-10 models), whereas the first PDP-8 model came out in 1965. OS/8 first appeared in 1971; I haven't been able to determine when its direct ancestor PS/8 was released. I ran some PS/8 programs under OS/8 V1, but I never ran PS/8 itself.
I appreciate the correction. Do you mean it was the first in the sense that "TOPS-10" was the new name for the PDP-6 monitor program as it was developed moving forward, and that all the PDP-7 and PDP-8 operating systems came long after the -6, or more literally that TOPS-10 itself is where the inspiration for OS/8 came from? Or do you not consider OS/8 an OS?
I have to say I'm not an expert on the DEC OSes, so I won't pretend to know the exact timeline, but the PDP-6 was not a very popular machine, only 23 or so sold. So I would assume it's really PDP-10 TOPS-10 that was the inspiration. OS/8 itself is from 1971 according to wikipedia, but its predecessor is PS/8. And that same article mentions how the interface was patterned after TOPS-10.
I don't think it's accurate to say, "the PDP-8 and -11 OSes would have been inspired by TOPS-10," except in the sense that they were also inspired by the IBM 360 and other systems — ideas did cross-pollinate, of course, but circumstances conspired to prevent especially strong cross-pollination in this case. The PDP-8 predates TOPS-10 by two years, so there is a substantial amount of software for the PDP-8 that could not have been inspired by TOPS-10 — including, I think, early versions of OS/8 — and RT-11 and RSX-11 followed OS/8 rather closely, especially at first. And, as I understand it, inside DEC, the small-systems folks (the PDP-7, -8, and -11) were organizationally separate from the large-systems folks (the -6 and -10), I think in geographically separate locations.