Why does a stream of water behave like this when aimed against a spoon or into a glass?

What you are seeing is the result of pressure waves moving upstream against the flow of the water.

Pressure waves in a fluid move at the speed of sound (because sound is a pressure wave) and in water this is around 1.5 km/sec. So while the water itself is moving downwards the pressure change caused by hitting an obstacle can propagate upstream as long as the speed of the water flow is less than 1.5 km/sec. If you inserted a pressure gauge in the water where it leaves the tap you would see the pressure increase when you put the spoon in the stream of water. This increase in the pressure in effect inflates the column of water, and this gives the effect you're seeing in the video.

To actually calculate the effect and predict its magnitude would be a formidable problem because even simple problems in fluid dynamics can be computationally intractable. The usual tactic is to give up and reach for a finite element analysis program and a large computer. I mention this to give myself an excuse for not explaining why you get the pulsing effect in your second video. It too will be due to pressure waves travelling upstream, but exactly why you get the oscillation I don't know.