Question: When I consider my own existence with respect to time I can imagine three possibilities: 

(1) **Time extends infinitely into the past.** In this case, how can the present, with me in it, exist, since there would be an infinite period of time that existed before the present. That seems illogical. 

(2) **Time had a starting point from which everything in our universe evolved to the present (t=1).** That possibility would require something (God) to start the clock ticking (t=0). But in this case we're back to square 1 with the problem of an infinite past. So the only possibility that makes sense to me is a third possibility 

(3) **Where time is some sort of abstraction in human consciousness that we need in order to relate cause and effect.**

Answer: 
I think there may be a 4th option which is closely related to your option (2). That is, time started at the big bang and has existed independently of our minds since then forming our Einstein-Minkowski 'time-cone'.

Of course, what came before the big bang, or if this is even a question that makes sense, is up for debate but there certainly would be no need to posit a God figure to make this work and so you could avoid your first cause infinite regress.
3 is correct. Time is a abstraction, a filing system used to arrange events and memories into a logical system of cause and effect.

Per relativity, space and time are not discrete entities, but a single construct called spacetime. Saying something is X km long is equal to saying it is Y seconds long. All units of time are based on and can be reduced to measures of distance.

Time is a mental construct used to make sense of movement. Movement produces the sensation and experience of time, not the other way around.