Welcome to FT.
Before jumping on those credit cards since you mentioned traveling internationally and domestically, which airline is that for? Chase and other banks offer bonus miles which you can take more advantage of in the beginning. Sapphire Preferred is still a good choice as it allows you to transfer those points to airlines plus you get 20% point redemption rate for travel rewards.
Yes, you do get extra points if you book through the Chase ultimate Rewards Mall. It varies depending on which site you go for (Travelocity, Expedia, etc.).
What do you mean you"usually" pay off your balance every month? You should pay it off in full every month. No point getting interest charges and not sure how, if any, it will affect your credit. I always pay in full.
That is correct. As long as you sign up for those quarterly bonuses, you can get an additional 4 points per dollar plus your original 1 point per dollar spent.
Depending on how long your credit history is and what's your FICO credit score, you might want to look into getting cards from Citi and American Express also.