Why does the Orthodox Church recognize some sacraments from priests in some other denominations/jurisdictions but not others? I would think that if a priest is legitimate to perform a baptism he would also be legitimate to perform other sacraments.
No. A heretic is not a legitimate minister for baptism. If an Orthodox took their child for baptism to, say, the Episcopal minister and not the Orthodox priest, the child would not have a legitimate baptism, would not be a member of the Orthodox Church, and his parents are excommunicated from the Orthodox Church.
If it is done out of ignorance (i.e. early immigrants to the New World witout a priest), the Lord's hand is not shortened that He cannot save, but the proper thing to do as soon as possible would be to receive chrismation to make whole what is deficient (i.e. the baptism). In fact, the Orthodox layman is, in the absence of an Orthodox priest, more a valid minister of baptism than the pope of Rome.
If done willfully, the doors of repentence and chrismation are always open.
I guess this is leading me down the road of asking what makes a priest a priest?
Preserving the Orthodox Faith he received in the laying on of hands from an Orthodox bishop who has been elevated to the Orthodox episcopacy by Orthodox bishops in communion with the Orthodox Church's diptychs.
A sacrament is Grace from God. Ordination is a sacrament. The Holy Spirit carries out the sacrament regardless of the sinful condition of the Priest. So what does the Priest matter?
Hebrews 5: 1 For every high priest taken from among men is appointed for men in things pertaining to God, that he may offer both gifts and sacrifices for sins. 2 He can have compassion on those who are ignorant and going astray, since he himself is also subject to weakness. 3 Because of this he is required as for the people, so also for himself, to offer sacrifices for sins. 4 And no man takes this honor to himself, but he who is called by God, just as Aaron was.
Luke 9:1 Then He called His twelve disciples together and gave them power and authority...2 He sent them to preach the kingdom of God....4 “Whatever house you enter, stay there, and from there depart. 5 And whoever will not receive you, when you go out of that city, shake off the very dust from your feet as a testimony against them.”
10: 1 After these things the Lord appointed seventy-two others also, and sent them two by two...16 He who hears you hears Me, he who rejects you rejects Me, and he who rejects Me rejects Him who sent Me.”
Can a sacrament occur without a Priest?
Only baptism, and even then it is imperfect (but can be perfected).
Can a Priest perform a sacrament which does not confer the Grace of God through the Holy Spirit?
Church tradition had the appointment of Bishops through a lottery as it was not important which member fulfilled the role.
Not according to Scripture, nor the Tradition of the Church:
Acts 1:15 And in those days Peter stood up in the midst of the disciples[c] (altogether the number of names was about a hundred and twenty), and said, 16 “Men and brethren, this Scripture had to be fulfilled, which the Holy Spirit spoke before by the mouth of David concerning Judas, who became a guide to those who arrested Jesus; 17 for he was numbered with us and obtained a part in this ministry.” 20 “For it is written in the Book of Psalms:
‘ Let his dwelling place be desolate,
And let no one live in it’;
‘ Let another take his bishoprick."
21 “Therefore, of these men who have accompanied us all the time that the Lord Jesus went in and out among us, 22 beginning from the baptism of John to that day when He was taken up from us, one of these must become a witness with us of His resurrection.”
23 And they proposed two: Joseph called Barsabas, who was surnamed Justus, and Matthias. 24 And they prayed and said, “You, O Lord, who know the hearts of all, show which of these two You have chosen 25 to take part in this ministry and apostleship from which Judas by transgression fell, that he might go to his own place.” 26 And they cast their lots, and the lot fell on Matthias. And he was numbered with the eleven apostles.
My current understanding is that it depends on the Faith of the individual just like touching the hem of Jesus' clothing.
Then your understanding is incorrect (and Protestant). Think of Uzzah touching the Ark. (II Kigdoms/Samuel 6:6-7). The power is there regardless of the Faith of the individual touching it. How it touches the individual, for life or for death, depends on the Faith of the individual.