1) When I load the cleared game save data, I'm able to re-play all the storyline missions, but there doesn't seem to be any bonus features or a harder difficulty setting for me to play in. I still have to play the extra missions, does finishing those unlocks anything?

2)There is one thing I didn't understand regarding to the plot, it probably was explained but I don't remember it, why did the 5 days time-loop started? What triggered the philosopher's stone to turn back time in the fifth day the first time?

I see, thanks for the information, I'll have to finish all extra missions next.

When you say Demon Lord, you are referring to Grimlet? His awakening made the stone activate?

Then you mean the archmage Calvaros. He wasn't a devil lord if I'm not wrong though.
So since the stone was Calvaros' treasure, his awakening activated the stone, which triggered a 5 days time-loop. But why was Lillet the only who managed to remember what had happened when the time looped?

Mmm, for your second question, Shadow, my guess is that perhaps the very first Lillet somehow found the philosopher's stone and the moment she touched it out of curiousity, she unintentionally created a time loop where she was trapped and became old after thousands of years. During those times, she watched her own self loop over and over from that boundary of time. Then one opportunity finally came, the other Lillet comes and meets her, and they unexpectedly switch for the first time. The new Lillet gets trapped while the old one goes out. Then it is later followed by switching on the third time, and finally the game which you're playing is probably the time the 4th Lillet feels persuaded by the third trapped Lillet to destroy the stone rather than to inspect it.

Remember, the 3rd Lillet said "I am the cause." It is possibility that Lillet touched the stone when she first found it before the time loop started.

A bit complicated, isn't it? ^.^ But I'm not 100% sure of this, it's just my theory I forumulated through listening to the conversation over and over. If we could ask the makers of the game, they hold the true answer.