Yes, it's normal- probably he is not used to the food you are offering (many shops don't feed their Bettas or only feed them a tiny amount, so even if they recommended the food they use, it might not be what he was used to at the breeder). Give him time.
However, I recommend strongly that you do not leave uneaten food in the tank. He doesn't really need more than 3 pellets per day- most people greatly overfeed. With my Bettas, I feed one pellet at a time, and they don't get the next one until they have eaten the first. If a pellet is left I will remove it- the only time this happened was when I switched food, and a day of no food was enough to convince him to eat the next day.
Yes, this is something you need to fix. Being sucked into the filter can rip fins- they are very delicate. I only use sponge filters in my Betta tanks for this reason, and my Bettas still often choose them as resting places (I often wonder if they enjoy the very gentle suction for some reason?).
If you have a HOB (hang on back) filter you can buy a filter intake sponge very cheaply on Amazon so it will be impossible for his fins to be pulled in, if it's another kind of filter, a web search for the name/type and "Betta safe" or "baffle" should give you instructions on how to fix it so it's safe for him.