Vv 6,7: Primary fulfillment to Hezekiah: already born (note present tenses). Compare parallels to Isaiah 22 -- a Hezekiah prophecy: "government, father, shoulder, David, throne" (Isa 22:21-23).
THE GOVERNMENT WILL BE ON HIS SHOULDERS: The government of Israel is here symbolized by one of its emblems: the royal robe fastened at the shoulder. In Judges 16:3,4: Samson possesses the gate of his enemy Dagon, carrying it on his shoulders to Hebron. [The gate, especially of one's enemies, also symbolizes governmental power (Ruth 4:1, and other refs].
When we come to the New Testament, reference to the royal shoulder suggests the shoulder of Jesus, that bore the cross (John 19:17).
WONDERFUL COUNSELOR, MIGHTY GOD, EVERLASTING FATHER, PRINCE OF PEACE: This is probably intended to be all one name, like Mahershalalhashbaz (Isa 8:1). Following the Soncino Bible, this could be read: "Pele-joez-gibbor-el-abi-ad-sar-shalom" = (perhaps) "The wonderful counselor, the mighty God, IS the everlasting Father OF the prince of peace". (Some of these same "names" are combined elsewhere: in Isa 28:29; 25:1, for examples, about God Himself.)
However, if it be insisted that every phrase be applied, separately, to some human being, then -- first of all -- it may be applied to Hezekiah (he is the initial fulfillment of the "child" in Isa 7:14, and the "little child" in Isa 11), and then secondarily (though, of course, most importantly) to Jesus...
COUNSELOR: Used by Micah in Virgin Birth prophecy (Mic 4:9). "Counsel" refers to the personal fellowship between God and His Son: Psa 16:7; Mar 1:35; Joh 6:15; 5:19,20; 8:28; 12:49; 14:10. True counsel, wisdom, only from God: Jer 10:23; Job 12:13; Isa 40:12-14; 11:2-4. Christ brought God's counsel (Deu 18:15-18), as "one having authority" (Mat 7:29). See also 1Co 1:30; Col 2:3; Zec 6:12.
MIGHTY GOD: "El Gibbor" = "mighty hero" (Moffatt). The Father has given His own Name to His Son as well (Phi 2:9; cp Psa 45:3,6; Heb 1:8). In general Scriptural terminology, one who acts on behalf of God may be called "God": (a) angels: Gen 16:13; 18:13; Exo 23:20,21; Hos 12:3,5; (b) men: Exo 22:28; 22:6; 21:8 (elohim); Psa 138:1; Joh 10:34 (cit Psa 82:1,6); (c) the Messiah: Isa 8:13,14; 61:1,6; 64:4; 65:16; Zec 12:10; Mal 3:1; Joh 20:28; Heb 1:8.
EVERLASTING FATHER: In Hebrew idiom, "father" may signify "master, leader, teacher" (ie, Isa 22:21; Gen 45:8; 2Ki 5:13; Jdg 18:19; Gen 4:20,21). So, literally, this phrase may mean "father of the age to come" (LXX).
PRINCE OF PEACE: As was Melchizedek (Gen 14; Heb 7). "The peace" -- or the "prince who brings peace" -- "when the Assyrian comes into the Land" (Mic 5:5) -- about Hezekiah first, before it is about Jesus. Cp the angels' testimony at the announcement of Jesus' birth: "Glory to God in the highest, and on earth PEACE to men on whom his favor rests" (Luk 2:14).
I hope you have found this helpful.