我最近没打过羽毛球 "I haven't played badminton recently": is the 过 necessary?
Your 过 is not necessary, but:

我最近没打过羽毛球。and 我最近没打羽毛球。

are equivalent, as far as people I asked are concerned. 

For the question: 'Have you played badminton recently?' people immediately used ‘打过’， but I did not consult all of China!

你最近打过羽毛球吗?

Apparently, ‘打了’ conjures up the idea of hitting someone, so would tend to be avoided in this situation.

I've never played badminton.
我从来没打过羽毛球。

I would like to ask 水巷孑蠻先生 (and other experts here), whether this use of 过 is found in Old Chinese and Middle Chinese texts, or whether might be a concession to Western ideas of tense and especially past tense. 

Any situation where a time in the past is mentioned: 'Yesterday .., Last night .., Recently ..' we know for certain the event mentioned is in the past. Past Tense is actually redundant. English sometimes uses what is quaintly called the 'Historical Present': 

"Last night, I'm sitting alone, watching TV, when, all of a sudden ....." Nobody will say, 'You used 'I am', therefore this is a Present Tense sentence.'