If a place is publicly funded, all or part, then to any degree does that make it a public place also?
No. There is a lot of tricky jurisprudence about what exactly a “public place” is. Even places that are wholly public funded don’t necessary meet that definition.
For example, the courthouse steps are probably 100% a “public place”, the inside of a courtroom is a different category with more restrictions, mostly up to the discretion of the presiding judge, and the inside of the judge’s chambers in that courtroom is entirely NOT a “public place”. Yet all of those are publicly owned and funded property in the same building!