Electronic – TC358743XBG HDMI to CSI-2 Board

1) Having it connected to a PC is not necessary, an eyefi SD card is a great idea but you already have wifi onboard, why would you need another one?

2) The power source is the computer. But again, you can battery power the board. I don't know where that 6AA batteries comes from but that's probably too much, the PC provides 5V through the micro usb, why would you need 9V? Depending on how much autonomy you need, how big the batteries can be and how much money you can spend an appropriate battery solution might (or might not) be found.

3) That just depends on what who's answering likes more, I've used the raspberry but maybe someone else likes arduino more.

Any Openwrt-capable hardware with USB and Wifi will work almost out of the box (you may need to build and flash custom image with mjpeg streamer), no development necessary. The lowest-cost platform is a5v11 (~US$8 shipped), plus your usb webcam of choice plus usb wall wart. It will look and feel like a linux machine streaming video from a webcam over tcp port.

Higher frame rates are possible with appropriate hardware -> https://forum.openwrt.org/viewtopic.php?pid=296414#p296414 Raspberry Pi running Raspbian will work out of the box too, but it's less cost effective and needs external wifi interface.