Are mainline Protestants our Brothers in Christ by virtue of the Trinitarian Baptism? [Catholic]

Although they might be validly baptized (anyone can baptize), they are not members of the Church because they do not believe the Catholic faith in its entirely.

As Pope Pius XII wrote in his encyclical Mystici Corporis Christi:

Actually only those are to be included as members of the Church who have

1. been baptized

and

2. profess the true faith,*

and

3. who have not been so unfortunate as to separate themselves from the unity of the Body, or been excluded by legitimate authority for grave faults committed.

“For in one spirit” says the Apostle, “were we all baptized into one Body, whether Jews or Gentiles, whether bond or free.” As therefore in the true Christian community there is only one Body, one Spirit, one Lord, and one Baptism, so there can be only one faith. And therefore if a man refuse to hear the Church let him be considered — so the Lord commands — as a heathen and a publican. It follows that those [who] are divided in faith or government [e.g., the Protestants] cannot be living in the unity of such a Body, nor can they be living the life of its one Divine Spirit.

^{*in its entirely; cf. Satis Cognitum}

Protestants are at least material heretics; they do not hold the same faith as Catholics. They deny the papacy, a truth necessary for being a member of the Church. They do not even recognize the visible Church (its government). And, as Pope Leo XIII said in Satis Cognitum, an encyclical on the unity of the Church consisting in being “one in faith, in government, and in communion:”

The practice of the Church has always been the same, as is shown by the unanimous teaching of the Fathers, who were wont to hold as outside Catholic communion, and alien to the Church, whoever would recede in the least degree from any point of doctrine proposed by her authoritative Magisterium.

^{adapted from this answer; cf. also this answer}

Certainly not.

If a protestant is culpable of his exclusion from the Catholic Church. Yes the sacrament is valid, but by culpably rejecting dogma, he can possibly bring upon himself excommunication and therefore rejects the salvation offered to him.

Excommunication does not mean damnation, but if culpably persisted it will lead to damnation.

I highlighted culpability, to emphasise only God knows the full culpability of a man's sin!

An example of the beliefs that can lead someone to excommunication are...

If any one saith, that, since Adam's sin, the free will of man is lost and extinguished; or, that it is a thing with only a name, yea a name without a reality, a figment, in fine, introduced into the Church by Satan; let him be anathema.

Council of Trent Session 6 Canon 5

If any one saith, that by faith alone the impious is justified; in such wise as to mean, that nothing else is required to co-operate in order to the obtaining the grace of Justification, and that it is not in any way necessary, that he be prepared and disposed by the movement of his own will; let him be anathema.

Council of Trent Session 6 Canon 9

If any one saith, that it is necessary for every one, for the obtaining the remission of sins, that he believe for certain, and without any wavering arising from his own infirmity and disposition, that his sins are forgiven him; let him be anathema.

Council of Trent Session 6 Canon 13

If any one saith, that baptism is free, that is, not necessary unto salvation; let him be anathema.

Council of Trent Session 7 Canon 5 on Baptism

If any one denieth, that, in the sacrament of the most holy Eucharist, are contained truly, really, and substantially, the body and blood together with the soul and divinity of our Lord Jesus Christ, and consequently the whole Christ; but saith that He is only therein as in a sign, or in figure, or virtue; let him be anathema.

Council of Trent Session 13 Canon 1

A few groups could be found guilty for at least one of these.

According to the official Catholic theology from Vatican II, the answer is YES. Here's a quote, I've cited before on a similar topic.

Wikipedia Quote)

J. M. R. Tillard goes into detail, in New Catholic Encyclopedia, about "the development of a carefully nuanced vocabulary, consistent with Vatican II Ecclesiology," which evolved from "the idea of membership in favor of that of incorporation" and has its categorization found in the dogmatic constitution Lumen gentium (LG) which Tillard describes: Catholics are defined as "'being incorporated' (incorporatio), qualifying the term with the adverb 'fully' (plene) and emphasizing that full incorporation requires the presence of the Holy Spirit."[b] Non-Catholics and catechumens are defined as "'being linked' (conjunctio) to the Church, again carefully stressing the role of the Holy Spirit in each case."[c] Non-Christians are defined as "'being related' (ordinantur), a term that suggests a dynamic relationship, an orientation toward the Church."[d] "Every shade of difference in meaning among these terms is important," emphasizes Tillard. "But the terms acquire their full force only in the light of the most authoritative commentaries on them," UR and Nostra aetate (NA). "Then, supposing the nuances indicated, the richness of such expressions as the following becomes clear: 'Churches and ecclesial communities';[e] 'separated brethren';[f] 'separated Churches and ecclesial communities';[g] 'full communion'—'imperfect communion'."[10][h] "But thanks to its ecclesiology," wrote Tillard, "Vatican II was able to affirm at the same time that Churches or ecclesial communities separated from the Catholic Church are part of the single Church, and that nevertheless incorporation in Christ and His Church possesses within the Catholic Church the fullness that it does not have elsewhere."

Unitatis redintegratio (Wikipedia)