download any video from the web to my Iphone

This is a somewhat difficult area, as this may technically be illegal in some cases...You can do a google search for sites that allow this action. BE CAREFUL as some of these sites have sketchy protocols...

However, one way is to take a screen recording of whatever video you want to save.

Here's a link showing you how to take a screen recording!

Record the screen on your iPhone, iPad or iPod touch – Apple Support (UK)

Hope this helps!

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Which in many cases won't work or you don't get sound.

Copyright is something that exists and many websites or devices adhere to the law.