Well I must say that it seems like you have attempted to do some research. To answer your question. Yes. You are able to leave the father with the fry and even for more than a week. Usually until the fry are 3.5-4 months old. At least it has been reported until then.
I am currently testing this method and read up about it here:
Please read that and then read this one:
Those are very good and are very helpful. Personally is would suggest if someone is going to try this method to have some practice spawns under their belt. Try the traditional method a few generations and then go ahead and try this. Over all make sure you research a lot more. Im glad you came here with some kind of knowledge about breeding bettas and have the willingness to do your research.
To answer your other question. No. Frozen brine shrimp or any other frozen foods cannot be used as a substitute for live foods when fry are still young.
I usually start using frozen and dry foods when my fry are 2-2.5 months old. They are usually .75-1 inch when I start feeding them this.
Live is most important in the fry's health and growth. I have heard of people using dry commercial bought fry food, but the reviews say that it never ends to the fry's full potential.
I hope you continue to research as well. There is much more to breeding than you may think.
You need jars to separate the males, grow out tanks/tubs, homes for the fry unless you plan to sell online, be prepared for everyday 40-60% water changes daily, ect.