Heyho and welcome to Tiny Core!
I may not be able to answer all of your questions, but some suggestions:
1. If you're using and developing much data, you could add the boot code home=sda1 to your boot loader kernel line, which will use a home folder on the first partition of the first hard disk recognized. No need to backup any things in your home folder then (which is basically /home/tc/).
2. To save extensions on your hard drive you should set the tce=sda1 boot code as well. This way downloaded extensions will be loaded from the disk.
3. If you want to make other settings persistent, e.g. setting an own password for user tc it's quite simple: sudo passwd tc, enter the new password, add etc/shadow and etc/passwd to /opt/.filetool.lst and run "Backup". The Backup procedure is necessary because the whole operating system runs in RAM, which is one of the advantages of Tiny Core. It will always be the same after a reboot unless you configure the backup to change anything
(btw. most of those topics are explained better in the wiki)
4. A "basic" Tiny Core installation doesn't consist of many files. You got the kernel (vmlinuz), the Init RAM disk (core.gz), some boot loader stuff (like grub/menu.lst e.g.) and additionally a tce/ folder with extensions, a home/ folder with your private data if you wish and a backup file called mydata.tgz. If you copy all that to your "installation" hard disk, it's quite the same like having it on an USB key and it will work too. So no need to start over and over again. You could start with an USB key, copy everything to the hard disk at some point and install the boot loader (this is indeed necessary to make the hard disk bootable).
5. For PHP you could use the php5.tcz or the apache2-mod-php5.tcz extensions, for C/C++ use the compiletc.tcz extension. Both should allow you to start developing immediately.
6. I don't have a clue about Verizon broadband, but it's probably circling around the topics PPP and USB serial. Try to search the forum, there were multiple entries and some solutions (basically it's like in every other Linux Distro in Tiny Core, you just need the proper extensions, e.g. the one that contains the pppd program or something like pppsetup).
7. @Out-of-the-box: That's right. Out-of-the-box means at least several hundred MBs installed on some real hard disk, because you have to support everything like Verizon broadband ;-). Tiny Core is a toolkit, it's not a turn key distro. Once you have all the "tools" (=extensions), you probably have the minimum necessary and also the fastest OS for your machine.
8. Forum-fixed-width: Could you tell me the browser and the OS you're using? It's not a special setting, on every browser I use (chrome, firefox, opera) it's reducing its width automatically to fit the screen. If anyone else is having this problem, please report.