Are weather changes periodic? (Do the same weather patterns repeat over years?) (Cycles) (Journal paper suggestion)

I'll address this part of the question first:

"Or a similar question can be: the weather measurements of how far ago can affect today's weather measurements? Does the weather forget about the weather events of long time ago? If so, how long is that window?"

The answer is "approximately two weeks". Even with nearly perfect knowledge of the atmospheric state (the distribution of pressure, moisture, aerosols, etc), the chaotic nature of fluids means that beyond two weeks, one would be better of predicting the weather from a climatological estimate (i.e, the average of the weather on that date for the last ~30 years). Research into this very question was foundational for the field of Chaos theory (Lorenz, 1963 is a famous example). This isn't just an artifact of the predictive models, but the natural system itself. The weather has effectively 'forgotten' what it was doing a few weeks before.

While the atmosphere has a short 'memory', other parts of the Earth have longer 'memories'. Notable examples are the Oceans and the Landsurface. A warm anomaly in the Ocean will stick around for a while, since ocean currents are much slower than air for mixing things around (and diffusing out the anomaly). The atmosphere is very influenced by what happens on the surface, so the warm/cool anomaly can affects the weather in various ways. This is what happens in the case of ENSO that you have mentioned. The land can also have an impact. The amount of saturation in the soil can have quite a big impact on weather, especially for things like thunderstorms and monsoon systems. Continental snow cover extent may also impact atmospheric predictability. Stratosphere/troposphere coupling mechanisms might also be a source of longer term predictability.

And this part: "I am wondering if there is long term (10 year, 7 year, 20 year) periodic patterns Similar to short term seasonal and periodic patterns in weather such as sea/land wind, monsoon, ENSO, ocean sea-saw, etc."

The most well known ~10 year climate oscillation is the Pacific Decadal Oscillation.