Hmmm.... curious. Lines 2 and 3 are fine. I would not say, "I played tennis lately". However, "I have not played tennis lately" and "I play a lot of tennis lately" is fine. So, How can I explain this? Recently refers to the recent past. 'To have done something recently' place that something in a narrow span of the recent past, but does not include the present. "Lately" translates more as 'nowadays', 'these days', 'currently'. It is an expanded present that includes the recent past but implies that the activity continues through the present. "Lately, I question the value of my time spent on italki.com", implies that this is both a recent concern as well as an ongoing concern. "I recently took up tennis. Lately, I have been playing a lot," implies that in the near past I took up tennis and that I continue to play a lot. In essence, 'recently' is situated firmly in the (recent) past while 'lately' includes the present as well. I hope that helps.

Are my sentences correct? No. I played tennis recently. Lately, we have had snow instead of freezing rain. I think this is a question of usage rather than grammar rules.