Hi, I'm not sure I can help but I'll share what I've done and heard throughout the years... firstly exercise effects people differently, some people go high after exercise, I always go low. Also, exercise can have the effect of making you go low only a few hours later.
Re lantus, my understanding is that lantus works strongest after you take it and then as you come to the end of the lantus period, it slows down. this may be explaining your evenings highs which is just before your next due dose. This is also why people split the insulin , like I did, to avoid these peak and fall times. Because as one dose trails off , the other one is still working fine, and so the cycle continues.
Additionally, if you find you go low in the night, splitting would help you be able to significantly lower your evening lantus dose whilst your morning one can stay like regular (or half of regular as the other half is going at night). Of course the morning one will have some affect at night still but not half as much.
Another consideration is to switch your insulin time from night to one time full dose in the morning. I've done this too, to avoid lows morning hours.
Good luck!