(...this time re: Adam, Christ, believers, faith, faithfulness, obedience, etc.; perhaps more lists to come re: covenant of works, Abraham, Moses, New Covenant and re: sacramental efficacy and apostasy; I welcome interaction!)
1) The obedience required of Adam in the covenant of works could only have resulted from faith in God's Word.
2) The obedience of faith required of Adam in the covenant of works differs in character from the obedience of faith referred to in Romans 1:5 and 16:26 and described in similar passages under different language in two respects: a) in that Adam was obligated to perfect obedience to fulfill the covenant of works, whereas one under the covenant of grace is obligated to evangelical obedience as evidence of saving faith-union with Christ; b) the object of faith for Adam was God's word of promise respecting his covenantal desert upon perfect obedience, whereas the object of faith for one under the covenant of grace is God's word of promise in the gospel.
3) Should Adam have fulfilled the covenant of works, he would have been obligated to thank God for enabling him to do so, not only at the baseline of having created and sustained him in being, and not only in exercising the special act of providence in entering into the covenant of works with him, but in providentially (by the Spirit) preserving him in obedience and favorably ordaining that he so succeed.
4) A believing member of the covenant of grace is obligated to thank God for entering into a covenant of grace, effectually calling him so as to unite him to Christ by Spirit-wrought faith, forgiving his sins and covering him in Christ's righteousness, and for preserving him in Christ unto the Last Day (among many other blessings).
5) The obliged thankfulness of the (theoretically) obedient Adam is for divine "grace" only insofar as that term is stipulated (or recognized biblically at times) to refer to free divine favor in a broad sense, and must be carefully distinguished from the "grace" experienced by believing members of the covenant of grace which is stipulated theologically/confessionally as sin-covering/non-sin-imputing and undeservedly righteousness-imputing.
6) The righteousness performed by the (theoretically) obedient Adam could have been said to "inhere" in him in one sense, although it would have needed to be considered a divine gift of favor in another sense. It would have "inhered" in him in that it would have been performed directly by him; it would have been divine gift in that it would not have been (and could not have been) fully autonomous, creaturely as Adam was, but rather empowered by the Spirit and the Father and Son's favorable providence.
7) The righteousness credited to the believing member of the covenant of grace cannot be said to "inhere" in him, because he does not perform it himself, except in the marginal sense that there is a relationship between the perfect righteousness of Christ credited to him at conversion and the measure of imperfect, evangelical obedience he performs in this life as the Spirit gradually conforms him to the image of the same Christ (a work fully completed morally at death and holistically at the resurrection).
8) The righteousness credited to the believing member of the covenant of grace must be said to be a gift of divine "grace" in the technical theological/confessional sense of demerited favor which covers sin, which would not have been true of the theoretically obedient Adam's righteousness.
9) The evangelical obedience performed by the believing member of the covenant of grace is also a gift of divine "grace" in the technical theological/confessional sense, because it is Spirit-wrought in the context of original spiritual death or "uncircumcision of the heart" which would not have needed to be overcome in the case of a theoretically obedient Adam.
10) The righteousness performed by Christ "inhered" in Him since He performed it directly, although we must acknowledge that the Father "gave" the Spirit without measure to His incarnate Son, in order to enable Him to fulfill His ministry as the Last Adam. According to His human nature, Christ could not act (nor live or exist at all) autonomously from the Father, much less fulfill the Messianic mission. According to His divine nature, He was (and is) autotheos and a se, and Messianic obedience properly "terminates" on the Person of the Son, yet, He never acts (according to either nature) in autonomy from Father or Spirit.
11) Despite the ontological and Messianic dependence of the incarnate Son upon the Father (and the Spirit), according to His human nature, and despite even His "personal dependence" upon the Father according to His divine nature (i.e. the Son eternally receives distinct divine personhood from the Father, while His divinity is underived/He is autotheos), it is probably not helpful to speak often of the righteousness of the incarnate Son as a "gift" from the Father in any sense without heavy qualification, lest it be confused with the "alien" gift-righteousness received by sinners, and lest it diminish the inherent obedience of Christ as a centerpiece of the accomplishment of redemption which He Himself achieved by His own worthiness.
12) The phrase "the Law" in Pauline epistles usually refers to the Mosaic Covenant as a whole, although often with at least a rhetorical emphasis on human performance of moral duties.
13) The Mosaic Covenant made provision through the sacrificial system for sins short of high-handed rebellion/apostasy/contumacy. Therefore while the Law of God ultimately obliges all to perfect moral obedience, the covenantal accommodation of the Mosaic Law to sinners in the era of the flesh obliged its members to faith in the typified and promised Messiah, and to a measure of evangelical obedience evidencing saving faith (although such obedience may not have been expected to be as extensive or mature as under the New Covenant). Therefore we can say that members of the Mosaic Covenant (as is true of those under the New Covenant as well) were obligated to saving faith for individual, eternal justification, and that saving faith always resulted in "covenant faithfulness" and a measure of evangelical obedience (to which we may say they were obligated as evidence of saving faith and even as an "instrument" of certain blessings [but not as an instrument of "justification" in the normal, technical, theological sense--faith alone is its instrument, and Christ its only ground]).
14) Christ as Last Adam fulfilled the covenant of works but did so as one fully under "the Law," i.e. the Mosaic Covenant. Therefore we can describe His Messianic obedience as covenant faithfulness under the Law as well as perfect righteousness fulfilling the essence of the covenant of works (even if the precise phenomenology of the covenant of works was slightly different in Adam's case--i.e., the coherent cluster of the following: the moral law written invisibly on the conscience only [cf. Rom. 1, 2], the duty to slay the serpent and so guard the holy temple-garden of Eden, and most uniquely the positive law proscribing eating of the tree of knowledge of good and evil).
15) It was necessary to Christ's obedience as Last Adam that He exercise faith in divine promises (e.g., Ps. 2:7-9; Is. 53:10-12), yet this promise-receiving faith differs from the faith exercised by sinners unto salvation in several respects: a) the promises believed are not regarding salvation from sin in His case; b) the faith exercised is not required because of any sinful spiritual lack (the only "lack" that could be spoken of at all is the natural weakness of the flesh according to His human nature); c) the faith exercised is perfect and issues in perfect obedience to the revealed will of the Father; d) the faith exercised is not the sole instrument of Christ's justification/vindication (rather, His entire obedience, inclusive of but extending far beyond simple belief in all of God's Word, is the ground and instrument of His own justification/vindication, and is the ground of the justification of sinners [with faith the sole instrument in their case]).