What is the difference between these sentences? (过 vs 了) What is the difference between:

你最近看过电视吗？
你最近看了电视吗？

and

你最近去过杂货店吗？
你最近去了杂货店吗？

And similarly, if I want to say &quot;recently I went to my favourite cinema and found it's closed down&quot; would I say 我最近去了我最喜欢的电影院，可是发现了它已经关闭了? Or should the first and second 了s be omitted? I know that this 了 means &quot;action completed&quot; but I've seen many example sentences where actions are completed yet there is no 了, which has left me confused about when I should include it.
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".