How can I see all test data in CiviCRM?
I am unaware of a way to turn on test data as you put it but here are some work arounds.

To find test the records, in searches (advanced search, membership search etc) there is a field "Membership is a Test" by default this is set to No but if you want to find test record such as the one specified above you can switched this to Yes.

Another option, if the reason you are using test mode is to test a payment processor without spending money and this site is just for testing then I would put in your test processor details into the live payments section for you payment processor. Then you can run "Live" actions but not spend money. You can always change these details back afterward.