What is the difference between these sentences? (过 vs 了)
In casual conversation without further context, sentences in the first two groups are almost the same.  If no additional information is available, both expressions are acceptable to a native speaker. But using "过" is still better.

"了" vs. "过" :

"你最近看过电视吗？" means "did you watch TV at least once recently."  The time, TV program and duration of the action "watching TV" is not being asked.

"你最近看了电视吗？" means "have you watched TV recently".  And "了" here implies that "watching TV" is finished/done.  It is used when talking about the consequence of the action.

The translation of the last sentence in your question ("recently I went to my favorite cinema and found it's closed down") is acceptable.  But it is better to omit that "了" after "发现".  

"发现了" is used when a noun is after "发现了". Example: "X发现了新大陆", "X found new continent".  But when a sentence with "了" is used after "发现了", then you can omit "了" in "发现了".