Please post separate items in different topics if you can, so they are not getting confused with each other.
With the first one, I cannot replicate this. As you can see here, there is no problem when I upload. Unless I am misunderstanding the issue?
With videos, it depends on the browser. The software itself simply posts these in video tags, and its the browser which decides if it can play them or not, based on the available software and codecs it has available to it.
If this is only missing in one browser, this can only really be cache, as from the point of view of the software its either present or not.
I would ask here first of all for you to check all these on a default unaltered theme, with all 3rd party items disabled, and any caching on your server also disabled. This does appear to be very specific to you and your site unfortunately.