Answer to Question 1
TRUE
Answer to Question 2
The core was almost all hydrogen to start with, but with fusion reactions the helium content of the core has increased. As a main-sequence star, it will continue to brighten slowly for the next five billion years, then when the core is dominated by helium, the core will contract and heat up, forcing the photosphere outward and swelling it up into a red giant, probably burning up the Earth in the process.