Per wiki policy, Spoilers Off applies here and all spoilers are unmarked. You Have Been Warned.
Takeshi's mom was Twilight's previous owner.
She is apparently hiding a few secrets from her family. Kazuma was Twilight's previous owner.
Takeshi is somehow connected to him. Based on what we see of him during the flashback in episode two, he also has the dichromatic eye thing going on, and they're somewhat similar in appearance.- Jossed. It was Violet.
Interpretations of Takeshi's dreams.
Some of them are obvious, such as his brother's betrayal, but other ones are unclear, like Kurumi and the sea of blood. Takeshi becomes Kazuma.
Kazuma has Isoshima's picture in the finale and heals Takeshi's mom. Also, both Gekkou and Takeshi seem to have travelled to the past via Timey-Wimey Ball.