No, you can't. I think you may have misunderstood the meaning of the word 'lately'. 'Lately' means the same as 'recently'. These two adverbs describe the period of from a short time ago until now. Similar phrases are 'for the last few days', 'in recent weeks', 'over the past couple of months' and so on. These expressions need to be used with the present perfect: 'Why have bad things been happening a lot lately?' This means that bad things started happening, say, a few months ago and they are still happening now. If you want to refer just to the present time, you'd use the present continuous: "Why are bad things happening at the moment?" NB Students from Asian countries often use 'recently' and 'lately' instead of 'at the moment' or 'these days'. As for the present simple, we use this to talk about general facts and repeated events. For example: "Why do bad things happen to good people?" or "Bad things always happen when you least expect them". I hope that helps.

Yes, you can use 'lately' with the present tense but not the present simple. I would use a different verb with your example sentence - why have bad things been happening a lot lately. It is the present perfect continuous tense.

If you say it, most people will not know it is wrong or right. It sounds okay. BUT...it is more common to say Why Are Bad Things Happening Lately? OR Why Do Bad Things Happen All The Time? Why do bad things happen? is a question about why bad things happen in general all the time. But the word "lately" is about a specific time. One is general and one is specific. So we shouldn't mix them.