Are priests ever said to belong to the "high priesthood", or only bishops?

The principle followed with respect to Priests from new Church that come to Tradition is this: (1) If the candidate himself has doubts about ordination that he would like to be resolved, and a (2) Traditional Catholic SSPX Bishop, after examining all the details of the case, decides and judges that conditional ordination is needed, then that is done.

Sometimes (1) As with Fr. Gregory Hesse, the Priest himself has no doubt. (2) A Traditional Catholic Bishop or Bishops examine the case and judge that conditional ordination, in this case, is not needed. And then the Priest simply begins offering the True Mass.

Laymen are not in the final analysis either capable or competent of passing judgment on such matters in such a way as to bind others to their personal opinion. If a layman has doubts, he has every right to seek out Traditional Catholic Bishops and Priests Consecrated or Ordained in the traditional rite. But not to presume to say that every Bishop in the new Church is invalid. That is practically heretical. As Fr. Marie has proven, it contradicts the Indefectibility and Apostolicity of the Roman Church for Her to be without valid Bishops. No wonder some who believe that false opinion, contradicting Bishops and Priests, are tempted to lapse into "Orthodox" schism.

When valid form and matter are used, presumption is for validity, not against it, unless some contrary intention is explicitly manifested in some special case. This is what will have to be judged later on. Telling Father RomanTheo he is not a valid Priest without knowing anything about the specific details of his case is a different kind of  presumption. It's almost like telling a married man he is not married.

Traditional Catholic Bishops will investigate carefully before deciding. Then, laity should abide by that judgment made by the Bishops.

When a person comes to Tradition, he will receive all the graces he had thus far been losing out on. The vast majority of Traditional Catholic Bishops and Priests have endorsed that study by Father Pierre Marie and apply its conclusion in practice. We should recognize their authority and do the same.

Edit: And yes, the term for High Priesthood that is used is summum sacerdotium in Latin. That can be verified in the SSPX study itself.

When valid form and matter are used, presumption is for validity, not against it,

Isn't this whole thread a debate on IF the form is valid?  That's why the sacrament cannot be presumed valid.

Straight up saying alleged priests aren't priests seems presumptuous to me.  I can see expressing doubt, but I don't see how anyone but the Church has the right to make that final judgment.  

(I personally don't think I have particular doubt, but its certainly possible I'm just wrong) 

Let's summarize what everyone agrees on.  A new rite of episcopal consecration was promulgated by Paul VI in 1968.  In the Apostolic Constitution promulgating the new rite he specified the essential form:

ACTS of  PAUL VI
 APOSTOLIC CONSTITUTION
 Approbation of the new rites for ordination of Deacons, Priests, and Bishops

Denique in Ordinatione Episcopi, materia est impositio manuum quae ab Episcopis consecrantibus, vel saltem a Consecratore principali, fit silentio super caput Electi ante precationem consecratoriam; forma autem constat verbis eiusdem precationis consecratoriae, quorum haec ad naturam rei pertinent, atque adeo ut actus valeat exiguntur: 

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Et nunc effunde super hunc Electum eam virtutem, quae a te est, Spiritum principalem, quem dedisti dilecto Filio Tuo Iesu Christo, quem Ipse donavit sanctis Apostolis, qui constituerunt Ecclesiam per singula loca, ut sanctuarium tuum, in gloriam et laudem indeficientem nominis tui.

Finally, in the ordination of a bishop, the matter is the laying of hands by the consecrating Bishops, or at least by the principal Consecrator, done in silence over the head of the Elect before the consecratory prayer; the form consists of the words of the consecratory prayer, of which the following belong to the essence, and are therefore required for validity.

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So now pour out upon this chosen one that power which is from you, the governing Spirit whom you gave to your beloved Son, Jesus Christ, given by the Same to the holy Apostles, who founded the Church in each place, to be your sanctuary in the unceasing glory and praise of your name.

Source: http://www.rosarychurch.net/consecration/paul_6.html

I think we all agree now that Spiritus principalis does not signify the power of the episcopacy.  Rather everyone including Fr Calderon, trad123, RomanTheo and many others all try to find the power of the episcopacy somewhere else in the rite.  But according to Catholic theological principles, both the Holy Ghost and the power of orders must be unequivocally specified in the ESSENTIAL FORM.  Even Paul VI implicitly concedes that point.  Therefore, at best, the NREC is a doubtful form and therefore all those consecrated according to this rite must be avoided.

Yes, the Novus Ordo apologists seem to be arguing that Paul VI screwed up (I'm sure it was an accident...not!) and misidentified the essential form and yet he still managed to give the Novus Ordo a valid sacrament despite the obvious intention to do exactly the same thing as the Anglicans and other "Reformers" did centuries ago which the Church has already ruled as invalid.  Yeah, go ahead and stake your salvation on that but when you end up in hell, you'll only have yourselves to blame.

In conferring the Sacraments (as also in the Consecration in Mass) it is never allowed to adopt a probable course of action as to validity and to abandon the safer course. The contrary was explicitly condemned by Pope Innocent XI. To do so would be a grievous sin against religion, namely an act of irreverence towards what Christ our Lord has instituted; it would be a grievous sin against charity, as the recipient would probably be deprived of the graces and effect of the Sacrament; it would be a grievous sin against justice, as the recipient has a right to valid Sacraments, whenever the minister, whether ex officio or not, undertakes to confer a Sacrament. In the necessary Sacraments there is no doubt about the triple sin; in Sacraments that are not necessary there will always be the grave sacrilege against religion.

Henry Davis, S.J., Moral and Pastoral Theology, London: Sheed & Ward, 1935, Volume III, page 27

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Various Errors on Moral Subjects
 Condemned by a degree of the Holy Office, March 4, 1679

1. It is not illicit in conferring sacraments to follow a probable opinion regarding the value of the sacrament, the safer opinion being abandoned, unless the law forbids it, convention or the danger of incurring grave harm. Therefore one should not make use of probable opinions only in conferring baptism, sacerdotal, or episcopal orders. (Denzinger 1151)

Innocent XI (1676-1689)

Source: http://www.rosarychurch.net/consecration/probable_priests.html