First, what exactly is DryOS-Shell, and what does it allow me to do?
It's the internal command line shell of the Canon operating system, sort of like windows command prompt, but much more limited. It generally doesn't do much useful, and you need to either physically connect to the camera UART (
https://chdk.fandom.com/wiki/UART) or run a special CHDK patch (discussed in this thread
https://chdk.setepontos.com/index.php?topic=11029.0) to access it.
Second, how do I dump my firmware rom? (PowerShot SX410 IS)
You can use the Canon basic dumper
https://chdk.fandom.com/wiki/Canon_Basic/Scripts/DumperThird, Are there any scripts to take a photo while the lens is still inside the camera? (On the PowerShot SX410 IS the lens is still inside the camera until you exit playback mode)
No. However, there are scripts which take photos while the mechanical shutter is closed, for example
https://chdk.fandom.com/wiki/Lua/Scripts:_Fixed_Exposure_Intervalometer and
https://chdk.fandom.com/wiki/Meteor_Intervalometer_with_Dark_Frame_ManagementFourth, is there a script that lets you zoom faster when recording? The zoom slows to a crawl for some reason while recording.
It is likely that you can zoom at full speed from script using the set_zoom_speed and set_zoom functions. I don't think you can change the zoom speed used when using the physical zoom controls. Full speed zoom would likely be picked up in the audio.
Fifth, is there a hardware mod that allows the PowerShot SX410 IS to charge through the USB cable instead of manually taking the battery out and putting it in the charger?
Short answer: No.
Longer answer: Technically, no doubt someone with enough electrical engineering skills could build their own charge controller and somehow integrate it, but it would be entirely from scratch. In general, only the newer cameras with micro USB connectors have charging hardware onboard.
Sixth, is there a way to setup and FTP server on my camera? Sometimes I want to add a script or two from my phone but can't since I don't have access to the SD card.
AFAIK sx410 does not support Wifi, and therefore has not network connectivity whatsoever, which rules out FTP.
You can transfer files over the USB cable using PTP, but to upload files, you probably need a custom PTP client which likely does not exist for your phone.
For Wifi enabled cameras:
Short answer: No.
Longer answer #1: Something like traditional network / socket functions are available, so a sufficiently skilled programmer could write an FTP server. Significant reverse engineering would be required to access the network stack.
Longer answer #2: Wifi enabled cameras communicate with a PC using PTP/IP. You can use the CHDK PTP extension over PTP/IP to transfer arbitrary files. However, you would need to CHDK PTP extension aware PTP/IP client on your phone, which does not exist.