Canon accessories does not work on all models.
Canon camera straps work great if you have a Canon camera.
The Powershot models will work fine for her and are affordable new, but you might find a good used Canon for sale in your local classified ads, local photo stores, or you might consider older models on clearance or refurbished at Canon's website, ebay or the discount online store, Overstock.com.
As you are choosing new accessories for your digital camera, you will find that you have the option between getting name brand accessories and generic versions. In most situations, it would be better if you chose the name-brand version. Digital camera accessories almost always work better if you purchase an accessory that was designed for your camera brand. The generic accessories will tend to work less effectively when coupled with a name-brand camera.
Is there any other cheaper speedlights that will work with the Canon powershot SX10I3 Digital Camera
Yes, with a FD to EOS mount adapter.
Yes it would, they are both integrated with the same system and are easily compatible. If you were to use a different digital camera it still works well but not the same degree as the Canon.
A recommended flash for the Canon T3i is the Canon 430EX II. This flash is about $270. However, honestly, any Canon flash will work well with the camera.
It is really your opinion, but let me analyze it for you. A used canon may save you money but be sure to check that all the functions still work. A new may be more expensive but with lots of new functions. So if you like to keep up with new technology buy the new canon camera.
he uses one of the canon vixia's (for the newer videos)
The canon A360 powershot cannot be used as a webcam. It was not intended or designed to be used that way.
The Canon EF won't work on a film SLR, unfortunately. It only works on DSLR cameras.
Check the battery, if you tried to use a different battery and it doesn't work then better have your camera diagnosed by a camera technician.