Order in salvation - Does having new life precede faith?

Does regeneration precede faith?
 
John 5:24 I tell you the truth, whoever hears my word and believes him who sent me has eternal life and will not be condemned; he has crossed over from death to life.
 
"He who believes has (past tense) crossed over from death to life."
 
And in the verses that seem to say the opposite, the verbs are almost always both present tense:
 
John 20:31  ... and that by believing you may have life in his name.
 
So I don't think the order is given here, our faith sustains our life, "the just will live by faith" may indicate this. And in some instances, I think "eternal life" refers to the future inheritance of the kingdom, not to regeneration, as here:
1 Timothy 1:16 ... for those who would believe on him and receive eternal life.
 
So this verse does not show an opposite order.
 
But what about this verse?
 
John 1:12 Yet to all who received him, to those who believed in his name, he gave the right to become children of God...
 
A right, or "authority," may imply a real choice, putting faith before becoming children of God, and thus before regeneration. But it may be that this authority is not to be exercised at man's discretion, as seems to have been the case in the sending of the disciples here:
 
Matthew 10:1 He called his twelve disciples to him and gave them authority to drive out evil spirits and to heal every disease and sickness.
 
But surely this does not mean they could heal just anyone they chose, rather this authority was to be exercised at God's discretion, instead of their own.
 
But if (as most seem to say) the order here is actually a logical one, and not an order in time, then this doesn't seem to matter so very much, especially since all these aspects are bound together, repentance and faith and new life are all required for conversion, and required to be united.