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: 
Is time a mental abstraction or does it exist outside consciousness?
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The latter. Time exists independently of us.

In all likelihood, time exists independently of one's consciousness. That is to say, while it may be that the experience which we refer to as the passage of time is in part created by our brains, there is still a physical process which it corresponds to, and this is something which cannot be suspended or initiated by our brains.

While it's possible that time only exists when there is something there to 'observe' it, time certainly still exists when people aren't around, as can be verified by simply using a video/tape recorder to record events in an area which is devoid of people or even all intelligent life (within a certain radius). Alternatively, there is any number of subtle signs in everything which suggest that physical processes proceed without direct human/animal observation.

Infinite timelines
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Does time stretch infinitely backward or have a starting point?

You say that either of these possibilities are less preferable to the option of time merely being an abstraction of the brain, and I understand this if you associate the idea of a non infinite timeline specifically with the idea of a deity. Also your option 1) (infinitely backwards timeline) involves the idea of infinity, which indeed may be internally contradictory in some way.

However, the possibility of time having a definite start point does not necessitate that there is some "entity" out there which initiated the timeline.

That being said, I don't know how to differentiate between option 1) and 2). All that we know is that the answer to 3) is that time proceeds without us, in some shape or form.
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.