To use the official one, since you have multiple partitions, I suggest you format the partition with Ext3. That makes it simple. If you are not familiar with partitioning and formatting, I suggest you download Parted Magic, to do it.
I use Grub legacy, not Grub4dos. This is similar to what I use.
title Tiny Core 3.5rc1
root (hd0,0)
kernel /tc3.5rc1/bzImage quiet opt=hda1 tce=hda1 home=hda1
initrd /tc3.5rc1/tinycore.gz
There may be some differences with grub4dos.
How does one set up persistence in frugal install on the official?
Primarily by using opt=hda1 tce=hda1 home=hda1 in Grub. There is also backup, which you can use for persistence in any directory other than tce, opt or home. Later, when you understand more, you can learn how to modify extensions instead of using backup.
To install applications (you need to be connected to the internet), start App Browser, click connect, then click on an application you want to install, and click Go.
To be easy to start, apps should be On Boot or On Demand. You can select this when installing, or make modifications later using Apps Audit.
I use:
gpicview for images.
epdfview for pdf files
xfburn for burning cds and dvds.
You can remove apps. So you can install several and see which you like.
There are some ideas for starting.
Don't be concerned about asking questions. There are helpful people here.
I think you will really like Tiny Core as you learn more about it.