I'm pretty sure you know about this, but for laymen who may not be familiar with the terminology.
The word "catholic" (little c) literally means "universal". And it is a strong doctrine of the "Catholic church" (big C) that the group of worshipers is singular and universal.
I guess Protestants have a harder time deciding who's a Christian; probably acceptance of the Nicean Creed is a popular standard, though I prefer "anyone who seeks to imitate Christ".
Rome absolutely accepts them as Christians.
It views them as wrong, but it does that for plenty of Christians that don't completely reject it's authority, too.
> The central point of Lutheran Protestantism is that it's open source: anybody with a Bible can decide for themselves how to implement it without asking for permission.
It would seem to me that the central point of Lutheran Protestantism is that there is a right way to implement Christianity, that right way is clear in Scripture, and having intermediaries between the believer and the Scripture is a barrier to the believer finding the truth in Scripture.
Lutheranism also seems generally pretty strongly defined by a particular opinion on what the right answers are, or at least a wide range of them, though somewhat internally divided by the degree of certainty ascribed to those answers.
Also I think you are right that many Protestants believe their opinions are clearly implicated by scripture, but it should be obvious on reflection that Scripture is not self-interpreting.
Strictly speaking, there is a distinction between "sponsor" and "witness". The sponsor must be a Catholic, the witnesses can be non-Catholic Christians. However, in my personal experience (having had one child baptised Catholic, and the other will be baptised soon), everyone just collectively treats the "sponsor" and "witnesses" as "godparents". The parish just tells people "one godparent must be a Catholic". The sponsor is just the first godparent listed on the baptism certificate.
As far as parents go, technically there is no requirement that either parent be Catholic, or even Christian. A Buddhist couple could ask for their child to be baptised as Catholic. Now, in practice, priests will be extremely hesitant about fulfilling such a request, but strictly speaking it is allowed.
One Catholic parent is the usual heuristic but it isn't actually the rule so isn't absolutely binding. For example, if a non-Catholic is known to regularly attend Catholic services, but for whatever reason is hesitating in formally converting, a priest may very well agree to such a parent's request to baptise their child.
From the doctrinal point of view, there are no non-Catholic Christians, just Catholics of various degrees of imperfection, is and everyone who intends to join Christianity, or to join someone else to it, intends joining to the Catholic Church as the two are identical and coextensive.
[0] it does recognize particular churches, but they are part of rather than separate from the universal Church; in addition to those in full communion with Rome, these include Christian communities who are out of Communion with Rome but who retain valid episcopal apostolic succession, including the various Eastern and Oriental Orthodox Churches, among others.
> Generally Protestants don't recognize the authority of the church at all (i.e. Sola scriptura).
Sola scriptura is a doctrine of a subset of protestantism; there are Protestants who don't adhere to it but instead to prima scriptura, or to doctrines that aren't clearly either. It is certainly not the case that Protestants as a whole reject ecclesiastical authority.