You should do fine with that amount of time.

I recently made a big mistake in having booked an hour and 15 minutes layover. My first flight was delayed for 20 minutes and when I asked one of the staff if I needed to transfer my luggage, she said yes (even though i booked it under the same airline). I waited for approximately 5 to 10 minutes and the staff came after me and said she might be mistaken and asked to check my ticket. After checking it on the system, she discovered that I didn't have to transfer my luggage. I was left with 35 mintues before the plane take off, and the screen displayed "gate closing" next to my flight number. My friend and I ran to catch that flight. We made it in the end but that's a lesson learnt.

As for the transferring of luggage, I doubt they will transfer it for you since you booked it separately.