Which Wolverine Existed During the Time Gap?

I think the movie answers all this pretty well, but let's look at your questions step by step:

When he wakes up at the end of the movie he remembers what he did in the past but has no memory of his new timeline.

This is true and is because it is his old-timeline version who awakes at exactly the time when he would awake from Kitty's control in the old timeline, just in the new timeline now. He remembers everything he did in the old timeline and in his past body and basically feels like the movie events from '73 were just a minute ago, as evident from his dialogue with Charles at the end

Charles: What is the last thing you remember.

Logan: I'm drowning.

But on the other hand he doesn't seem to have any knowledge of the events of the new timeline after his past self took control, since he asks Charles for a "history lesson" for everything after 1973, to which Charles just says: "Welcome back", realizing that the Logan he met 50 years ago has finally returned.

If waking up in the present caused his younger mind to take back control of his body would he have any memory of the past few days where the older mind had been in control?

I don't think so. The past self is more-or-less unconcious while his body is "posessed" by future Logan and he doesn't remember anything from that time when future Logan leaves again. This is evident from the scene during Mystique's assassination attempt on Trask. When Wolverine sees Stryker and remembers his traumatic experiences with him, he gets very "uneasy" and causes some trouble in Kitty's tries to keep the connection between future and past. Thus for a short period of time the past Logan is actually taking control of his body again (and the future self seems to be in some kind of limbo between the times/bodies) and doesn't have any knowledge where he is and how he got there (and needs to be convinced by Charles that he was on a bad acid-trip).

So when past Logan continues control after the events of the movie in '73 he doesn't know anything about his experiences with Charles, Erik and all the others (and it is unclear how he ended up with the X-Men, given that he was recovered from the Potomac by Mystique and would need to have been recruited somehow by Charles, similar to X-Men 1).

If his younger mind had taken back control once present Wolverine woke up when time caught up to the present wouldn't the younger mind be killed off?

From the above it seems so indeed. In fact there can only be one conciousness existing in Logan's body and it's clear that it is the old timeline-Logan who awakes in X-manor.

Lastly if the present mind lived out the difference in time and had the knowledge he had gained through the years what effect would that have on the outcome of future?

Hard to say, but since the present mind (if you mean the old-timeline Logan with this) is not living through the time difference in the new timeline, as explained above, this is entirely speculative to say. But he could certainly have used that knowledge in some way or another, sure, but nobody knows the outcome of this, maybe an even different future to the one we saw, but who knows.