How long would it take to travel 100 million light years?

Some galaxies will have fallen over the cosmic horizon, where no amount of time would ever let you reach them. If you wanted to travel 100 trillion light years away, you could make the journey in 62 years.

How old is the universe thought to be?

approximately 13.8 billion years old
Using data from the Planck space observatory, they found the universe to be approximately 13.8 billion years old.

What would happen if a telescope is not large enough?

If the telescope isn’t large enough, the resolutionwould be low, and the observer wouldn’t be able to make out things far away. To get an idea, the size (diameter of mirror for instance) of the telescope that’ll enable you to see an object of about 10m at that distance is about $10^{8}$km while the diameter of earth is about 12800 km. But again]

How far back in the past can you see the Earth?

If you had a mirror on the moon (about 238K miles away), you could see about 2.5 seconds into earth’s past. If that mirror was on Pluto, you could see about 13.4 hours into Earth’s past. If you are relying on hearing, you hear an event at 30 m away about 0.1 s after it occurs.

How old would I be on Planet X if I’m on Earth?

You look back at Earth – You are non-existent on Earth. Because Earth’s time does not bend back. It took 27 years for Earth’s image to reach you at the time you left. So unless you were at both planets at the same time you would be 27 on Planet X while on Earth you would be 54 Years old.

Can You See Yourself being born 27 light years away?

That is why runners often watch the starting pistol at an event, because they can see a more recent picture of the past than they can hear. To more directly answer the intent of your question: Yes, if you could magically be transported 27 lightyears away, or had a mirror strategically placed 13.5 lightyears away, you could see yourself being born.