What Happened to Rodolphus Lestrange (Bellatrix's Husband)?

Did Tonks kill him?

Tonks is a trained Auror, and her estimate (that she only injured Rodolphus) should be relatively good: in any case, it indicates she was not using a fatal curse. Unless Rodolphus was knocked unconscious or lost his wand, even a fall should not have been fatal to him. And the Death Eaters seem to have been trying to keep their injured from falling to their deaths, as well.

Did he abandon his wife and escape Voldemort like Karkaroff?

What Rodolphus thinks of his wife is anyone's guess--certainly she was in love with Voldemort, not Rodolphus. But it was their mutual loyalty to Voldemort that brought them together in the first place. Rodolphus was an old school friend of Voldemort's, as Slughorn's memory in Half-Blood Prince indicates, so his loyalty was probably personal as well as political. Rodolphus's actions after the end of the First Wizarding War are the best indication of his feelings in that regard. Voldemort had disappeared, but instead of breathing a sigh of relief and going on with his life (as Karkaroff did), Rodolphus joined his wife in attacking the Longbottoms. Yes, Bellatrix was probably the instigator, but other people seem to have perceived them as a team, and they went to prison together. Most people in the wizarding world consider Rodolphus a particularly hardcore Voldemort supporter. For him to attempt an escape at a time when Voldemort's power was greater than ever before would not make sense, and it would be inconsistent with his previous behavior.

Why wasn't he in the Battle of Hogwarts? Did J.K. Rowling forget about him?

Out-of-universe answer: Yes, I think there's a good probability that J.K. Rowling simply forgot about him. Many of the Death Eaters are essentially interchangeable (e.g., who cares what happens to Avery?). Rowling never gave Rodolphus much of a personality (unlike his wife and in-laws), so it is at least understandable that he disappears.

In-universe answer: There are a few possibilities.

1. He was at the Battle of Hogwarts, but he wasn't put into the same parts of the battle as Bellatrix was. Maybe his fighting skills aren't quite as good. Maybe he was needed to direct some other part of the battle. Maybe he was injured or killed early in the battle (Neville's Mandrakes come to mind), and so he isn't around for the fighting inside Hogwarts, which is when most of the other important Death Eaters are mentioned by name.
2. At the time of the battle, he is in some other part of Britain--or even in another part of the world--completing an assignment for Voldemort. When Voldemort calls the Death Eaters to Hogwarts, he can't come quickly enough to join in.
3. Rodolphus's injury was of a long-term or even permanent nature, rendering him unable to fight.

Hopefully, JKR will develop an in-universe explanation in the future.