My son will be flying into Copenhagen in July arriving at 7:15am. He will need to travel by train to Kalmar. He must go through border control. I am wondering: how much time he will need to get through border control at the Copenhagen airport and then onto the train?
Customs is usually a matter of seconds, unless he has something to declare or gets tapped for a random inspection, which is very rare.
I suspect you're actually thinking about immigration checks. This depends in part of what your son's nationality is, in particular whether he is an EU/EEA citizen. Could be anything from a few minutes in line to half an hour, depending on whether he's first or last out of the plane, and which of the two flights from USA that are scheduled to arrive at 07:15 get to their gate first -- plus whatever the risk of getting pulled out for a secondary interview is given his nationality and other circumstances.
However, ultimately it doesn't matter a lot. There's one train to Kalmar at 07:22, which he is not going to make it to in any circumstances. The next is at 08:22 on weekdays (arriving in Kalmar at 11:59), which he will make if his inbound flight is on time, unless something goes so wrong that all bets are off anyway.
(The next trains to Kalmar are at 09:22, every day, and 11:22, not Saturday).
Beware that these trains are comparatively often delayed, so do not make any rigid plans for when he will arrive in Kalmar.
As Henning has already mentioned, at that time of day, customs and immigration are usually fast.
There is a direct train to Kalmar once every one or two hours (8.22, 9.22, 11.22). This may be easiest since you don't have to change trains along the way. You can buy tickets in the airport at a machine. However, lines at the machines can be really long and they are not particularly easy to use. It may be better to buy the train ticket upfront at https://www.sj.se/en/home.html . The tickets are typically good for entire day, so if you miss the 8.22, you can still take the 9.22.
Example for tickets in July are here https://www.sj.se/en/home.html#/tidtabell/Copenhagen%2520Airport/Kalmar%2520C/enkel/avgang/20180705-0800/avgang/20180705-1500/VU--///0//
The train stops between Denmark & Sweden and technically it's possible that there will be another passport control on the train. That's pretty rare though.