Yes, however, sometimes the terms were used interchangeably.
Not of importance here.
Yes, the Sinaitic covenant was based on the law, and the law was based on the Aaronic priesthood
(Heb 7:11).
Why is this passage concerned with genetics? What part of DNA has anything to do with the Ten Commandments? The Law being referred to in this passage...
Heb 7:11 If therefore perfection were by the Levitical priesthood, (for under it the people received the law,) what further need was there that another priest should rise after the order of Melchisedec, and not be called after
the order of Aaron?
Heb 7:12 For the
priesthood being changed, there is made of necessity a
change also of the law.
Heb 7:13 For he of whom these things are spoken pertaineth
to another tribe, of which no man gave attendance at the altar.
Heb 7:14 For it is evident that our Lord
sprang out of Juda; of which tribe Moses spake nothing
concerning priesthood.
Here is the Law that this passage is speaking to...
Num 18:1 And the LORD said unto Aaron, Thou and thy sons and thy father's house with thee shall bear the iniquity of the sanctuary: and thou and thy sons with thee shall bear the iniquity of your priesthood.
Num 18:2 And thy brethren also of the tribe of Levi, the tribe of thy father, bring thou with thee, that they may be joined unto thee, and minister unto thee: but thou and thy sons with thee shall minister before the tabernacle of witness.
Num 18:3 And they shall keep thy charge, and the charge of all the tabernacle: only they shall not come nigh the vessels of the sanctuary and the altar, that neither they, nor ye also, die.
Num 18:4 And they shall be joined unto thee, and keep the charge of the tabernacle of the congregation, for all the service of the tabernacle: and a stranger shall not come nigh unto you.
Num 18:5 And ye shall keep the charge of the sanctuary, and the charge of the altar: that there be no wrath any more upon the children of Israel.
Num 18:6 And I, behold, I have taken your brethren the Levites from among the children of Israel: to you they are given as a gift for the LORD, to do the service of the tabernacle of the congregation.
Num 18:7 Therefore thou and thy sons with thee shall keep your priest's office for every thing of the altar, and within the vail; and ye shall serve: I have given your priest's office unto you as a service of gift: and the stranger that cometh nigh shall be put to death.
Num 18:8 And the LORD spake unto Aaron, Behold, I also have given thee the charge of mine heave offerings of all the hallowed things of the children of Israel; unto thee have I given them by reason of the anointing, and to thy sons, by an ordinance for ever.
Notice it is about DNA? Same as the passage in Hebrews. It is about the fact that under the Old Covenant, ONLY a direct descendant of AARON could be a Priest and yet Jesus Christ was a Jew from teh tribe of Judah. Because God will NOT COMPROMISE in the least with His Law, the Law had to be changed for Christ to become the High Priest after teh order of Melchisidec...
Heb 7:15 And it is yet far more evident: for that after the similitude of Melchisedec there ariseth another priest,
Heb 7:16 Who is made, not after the law of a carnal commandment, but after the power of an endless life.
Heb 7:17 For he testifieth, Thou art a priest for ever after the order of Melchisedec.
Heb 7:18 For there is verily a disannulling of the commandment going before for the weakness and unprofitableness thereof.
The disannulling of the Commandment that only Levites, specifically the sons of Aaron, could be Priests.
So when God replaced the Aaronic priesthood with the priesthood in the order of Melchizedek,
and made Christ the eternal High Priest of the new order (Heb 7:17, 21),
the law no longer had a basis, someone to administer it, and so was changed (Heb 7:12), set aside
(Heb 7:18-19).
The Law that determined the Priesthood by DNA was not even set aside, it was changed. Changed in that the High Priest now must be of the order of Melchisidec.
Therefore, the Mosaic law, which contained a curse for imperfect obedience (Dt 27:26; Gal 3:10),
has been replaced with the law of Christ (Mt 22:37-39; 1Co 9:20-21; Gal 6:2),
which is the law of love that fulfills the whole law (Mt 22:40; Ro 13:8-10),
and has no curse attached--a huge difference for the people of God.
The people of God are now under the law of Christ (Mt 22:37-39; 1Co 9:21; Gal 6:2), rather than the law of Moses (Heb 7:18-19).
When one reads the entire passage in context and then goes back and actually reads which Law was changed, it becomes obvious that Christ really told the truth when He said...
Mat 5:17 Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil.
Mat 5:18 For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled.
Mat 5:19 Whosoever therefore shall break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven.
Luk 16:17 And it is easier for heaven and earth to pass, than one tittle of the law to fail.
You know what? I actually believe Christ.