As to your first question:
Is 2 Tim. 1:6 a parallel to 1 Tim. 4:14? That is, do they refer to the same "gift" and the same "laying on of hands" event?
According to the several Bibles and commentaries I've examined, yes.
As to your second question:
What "spiritual gift" is Paul referring to in 1 Tim. 4:14?
Ignatius (Epistle to the Ephesians 13, ca. 107 CE) apparently believed that spiritual gift meant being:
- a follower of "the love of God towards man" and
- "stirring up yourselves by the blood of Christ".
Codices Sinaiticus and Alexandrinus, and Greek New Testaments (GNT's) N-A27, and N-A28 identify that spiritual gift as ΧΑΡΙΣΜΑΤΟΣ (χαρισματος / charismatos at 1 Tim. 4:14) and ΧΑΡΙΣΜΑ (χαρισμα / charisma at 2 Tim. 1:6) with both words referring to a divinely-conferred and free gift.
Zodhiates (Hebrew-Greek Key Word Study Bible 1984) explains, in part:
"The gift is called charisma [and is] the result of grace, charis [and] is exactly the same word [seen] in I Cor. 12 where gifts (charismata) are enumerated.... The same gift is referred to in I Tim. 4:14."
The New Living Translation (NLT) Study Bible also identifies the spiritual gift mentioned by Paul as charisma, and points out that it is:
"the Holy Spirit's enablement for ministry."
And as to your third question: "is Paul referring to ... the baptism in the Spirit and a gift that came as a result, or ... a subsequent event to the initial impartation/baptism of the Spirit?"
"In Act 13:2., when Barnabas and Saul were formally set apart to the mission campaign ..., there was the call of the Spirit and the laying on of hands with prayer. Here again meta does not express instrument or means, but merely accompaniment. In 2Tim 1:6 Paul speaks only of his own laying on of hands, but the rest of the presbytery no doubt did so at the same time and the reference is to this incident" (Robertson Word Pictures in the NT, where the laying on of hands by both Paul and the elders was apparently accompanied by prophecies uttered by the participating elders of the congregation; cp. Vincent Word Studies).