Specifically, am I allowed to travel on sidewalks on a university when I want, as I would be able to do outside of the university (which I know to be public property). If the state in question does not have a "stop and identify" law, am I required to provide identification if being detained on a university sidewalk?
You have two questions going here. One is "are sidewalks on (state) universities public property", and the other is "when do I have to provide identification in a state with no stop and identify law". I will focus on the second question. As it happens, property ownership is not relevant to that determination. The universal fact is that only a peace officer (variously identified) can force you to identify yourself, no matter where you are. In Ohio, which has such a law, the law requires you to give name, DOB and address if they suspect you and you are in a public place. Being in Kroger is being in a public place, even though Kroger is private property. Alabama likewise allows officers to stop and question a suspect in a public place. Arizona doesn't even limit the requirement to "public places". In Illinois, you will not be convicted of obstructing. That is because 725 ILCS 5/107-14 a peace officer "may stop any person in a public place" and "may demand the name and address of the person and an explanation of his actions". However, they forgot to include a part where the suspect has to answer in the statute. As far as I know, Washington state has no law allowing such questioning and again as far as I know you can refuse to respond. So: if you are a suspect, and the state has no stop and identify law, they cannot require you to identify yourself, and then in some states, you would even need to be in a public place.
And then there is Arkansas, where it looks like failure to identify can be taken as evidence of loitering (one would have to read the Arkansas case law to see if it's really that bad).
Publicly owned <> Public Space
All land is owned by someone. If the owner of the land is a government then the land is "publicly owned" otherwise it is "privately owned". Further, that land may be leased to someone else, in which case the controller of the land may be different from the owner of the land.
A state university is not an organ of the state government; they merely source (some) of their funding from the state but are legally independent, usually structured as trusts or public corporations. In that sense the land they own is "privately owned".
Notwithstanding, whether an area is a public space is totally independent of who owns it. The White House is publicly owned but it is most certainly not a public space. In contrast, a shopping mall is (generally) privately owned but the publicly accessible spaces are public spaces while the mall is open.
You are allowed to walk on the university's sidewalks when and how the university decides you can. If they decide to allow members of the public to do so then those areas are public spaces, if instead they erect fences and gates where you must swipe your student or staff ID to enter, they are not public spaces.