Yes, regeneration/quickening preceeds faith, which is a fruit of same, and it also is how justification is applied to the elect. Do you believe that the Spirit through regeneration causes all the elect to believe the gospel, wherein salvation is conditioned on the work of Christ alone? From your posts I gather that you believe that the regenerate elect can be ignorant about the righteousness of God, revealed in the gospel. This must mean then that you believe that an elect can believe in a false gospel, and yet be regenerate. Compare this to your statements of the regenerate having the fruit of the Spirit and it makes no sense.
From the eternal justification point of view I also wonder how it can make sense to say that all of the elect comes to faith in Christ, if such faith in itself is not indicative of being saved and unbelief is not indicative of being lost. I mean, if you are righteous/justified without personal faith, what good is such a faith to give evidence that you are an elect and what does such faith include? Is believing a false gospel, where salvation is conditioned on the work of the sinner, a sign of regeneration? An interest in biblical issues is enough? A "zeal of God", but no knowledge of the righteousness of God, revealed in the gospel, is yet a sign of regeneration? It makes no sense.