Yes, that's a very common arrangement with highways routed along surface streets in the Metro Area. I know with reasonable certainty that OR-10 west of Murray Blvd, all the way to the terminus at OR-219 is under the control of the Washington County Department of Land Use and Transportation. The county completely funded the widening projects between Murray and Kinnaman that occurred between the mid-1990s and now. I believe the section between there and OR-217 is under City of Beaverton control.
Almost: there is a stretch of Oregon 10 (Farmington Road) between MP 5.88 and MP 7.61 (roughly S.W. 198th Avenue east to S.W. Kinnaman Road), and from MP 8.68 east to an unknown point near Beaverton City Limits, that remains under ODOT jurisdiction.
The western portion makes sense as it is un-upgraded and I'm sure Washington County wants to see it widened or at least otherwise upgraded to modern spec before it'll accept jurisdiction. I don't get the eastern part since it's already upgraded.
I also am not sure on this, but at the far western end of Farmington Road at the intersection with Highway 219 for about one mile, it appears that as part of the recent intersection project that portion of road was built to ODOT spec including ODOT spec signage...not sure if it's simply because ODOT funded the work or if that last mile was turned back over to ODOT.
OR-210 (SW Scholls Ferry Road) is also almost entirely under Washington County control. OR-8 in Forest Grove, between OR-47 and B Street (the old alignment of OR-47) is under City of Forest Grove control, and the section west of B Street out to OR-6 is under county control as well.
Exactly - the only portion of 210 still under ODOT control is the portion from the P&W Railroad, northeasterly over Highway 217 to the intersection with Hall Boulevard (Oregon 141) - specifically, between MP 9.03 to MP 9.61, so just over a half mile. It still retains the ODOT route number of 143 for this tiny island.
Washington County paid for the Forest Grove bypass and then turned it over to the state; in exchange the old routing of Highway 47 in Forest Grove was turned over to the city. This results in the unusual situation of being at the intersection of Pacific and Quince Avenues in Forest Grove - with Oregon 47 passing through north-south and Oregon 8 passing through west-east, but the Tualatin Valley Highway (Highway 29) runs south-east, and the Nehalem Highway (Highway 102) starts and proceeds north. Not that many people travel north-south on Highway 47 through Forest Grove but for the motorist this has the unfortunate consequence of having a non-consecutive milepost while following "the same highway" - going from milepost 90 to milepost 18.
My understanding is that while Gales Creek Road has for a very long time carried Oregon 8, it never was an actual state highway...
One of the best ways to tell who controls what road is by looking at the fonts, size and text alignment on the "limitless" speed signs, believe it or not. ODOT's speed signs have a distinctive style that the counties and cities generally don't use.
ODOT seems to have several different styles of speed signs but they are definitely unique and generally not found on non-ODOT roads, unless part of an ODOT funded project. Washington County seems to have their own unique signs (in particular oversized numerals that are wider than the word "Speed", and older signs use a somewhat unique typeface for the numerals - in particular the '3's). Another tip-off are the "Adopt-a-Road"/"Adopt-a-Highway" signs, mileposts (non-existent on most non-ODOT roads), and any guide sign. Also, Washington County generally does not use warning signs larger than 24x24, while ODOT has no problem using a larger sign when needed, and until recently ODOT only used wood signposts while Washington County was the forerunner in using the square steel posts. ODOT recently has started using the square posts, and even some steel breakaway posts on non-Interstate roads.