This abandoned boundary import proposal came with written confirmation from ODOT that all TIMS layers are “available for public use, and no copyright applies”. I take that to mean they consider it to be in the public domain. ODOT has a more generous interpretation of the state’s sunshine laws than I’ve seen from local governments in the past, but I’m not one to complain.
Most of the occurrences of short_name and shortest_name that you’ve encountered in the Columbus area are a very old experiment that predates widespread use of short_name in OSM. Routine abbreviations went in short_name and the most important part of the name went in shortest_name (typically corresponding to tiger:name_base).
As I recall, it was an attempt to preserve distinctions that were lost after the U.S. community reached a consensus to unabbreviate words in TIGER-imported road names. However, the short_name and shortest_name tags appear indiscriminately, not just on roadways that would otherwise have ambiguous names. This experiment never gained traction anywhere beyond Central Ohio and somewhere in Wisconsin. I don’t think anyone is keen on preserving it at this point.
There is a newer, alternative approach to clarifying the ambiguous names: name:etymology with name:etymology:wikidata groups and elaborates upon the words that form the main part of the name. For example, in the Cincinnati area, there’s a West North Bend Road that we’ve tagged with name:etymology=North Bend, referring to the village by this name. An etymology would mainly be for the mapper’s edification, but a geocoder could have a heuristic that considers it when tokenizing the name.
Yes, the road-related layers in TIMS are generally reliable along state highways, but some of them are significantly less reliable along other roads. Ohio state law distinguishes between state-maintained state highways, which only exist outside corporation limits, and state-designated state routes, which continue through a city or village on locally maintained streets. Counties are required to certify county, township, and municipal road mileages with the state annually, but I don’t think this certification includes secondary attributes like speed limits.
Other TIMS layers have important caveats too: the proposed boundary import was abandoned in part because the “township” boundaries in TIMS correspond to historical or theoretical boundaries that would be accurate if cities and villages could never withdraw from a township.