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I am interested in whether (and if yes, how?) you would go about contacting an employee on sick leave in this very specific situation:

There are three people involved: CEO Carl, Manager Mike and Employee Eric.

Eric calls in sick on Monday. Mike and Carl have good reason to believe Eric is not really sick but rather venting off anger originating from a dispute with Carl.

Eric is scheduled to attend a conference on Tuesday. Eric did not mention whether he'll be back on Tuesday to attend the conference. So Carl and Mike have to assume that he'll not attend and thus have a conference ticket to spare.

Before Carl and Mike give the conference ticket to someone else: do they contact Eric to find out whether he'll be attending the conference? Note that this conference is one of the yearly highlights for Eric; Mike is aware of that.

Specifically in German legislation: how would they have to phrase the question so that they don't get in conflict with the law?

So Carl and Mike have to assume that he'll not attend and thus have a conference ticket to spare.

Don't assume, ask.

Eric already called in sick, so try contacting him by that same mean. Consider giving him a call in case there is a chance he won't see his email.

Now, being written or spoken, try something in the lines of:

Hello Eric. Sorry to hear you feel sick, I hope you get well soon. Just remember that tomorrow is the conference, so tell us if you don't think you will make it.

IANAL and I am not German.

If Mike and/or Carl have Eric's contact information, then they could just call him and ask: "Hey, we heard you're sick, are you going to make it to the conference?" In my estimation, the "we heard that" does not violate privacy laws; if a person is away from the office on unscheduled vacation, the common thing to hear (at least in my experience) is that the person is sick. Using company-usual contact methods (company email, internal chat, etc) is preferable if it is reasonable to believe that Eric will access those things while sick.

If Carl and Mike make a reasonable effort to contact Eric, and there is no response, then (imo) Carl and Mike can give away Eric's ticket. But first Carl and Mike should make their best good-faith effort to contact Eric to check if his illness is going to take him out of commission the day of the conference (sometimes you just don't feel well one day and want to skip work but are not actually "sick"; just because Eric comes in the next day perfectly healthy does not mean he lied about taking a sick day).

Specifically in German legislation: how would they have to phrase the question so that they don't get in conflict with the law?

There is no problem with the law no matter how you phrase it. Even a direct "I don't believe you are sick, please bring your doctor's note on Tuesday" would not break any law (assuming the employee has a contract that requires him to bring a note or you have sufficient indication that he might be faking it). For any non-German reading this, please remember Germany has no concept of "sick day allowance". You are sick when a doctor says you're sick. Going to the doctor and getting such a note is free (well, already included in your mandatory health insurance). And the employer is allowed to ask for proof in form of a doctors note, most put it into their contracts.

Please note that "strictly speaking not illegal" might not be the level you want to communicate with employees. Personally, I would feel very offended if someone thought I'd be faking it. So being nice and respectful certainly does not hurt.

You may also not want to disturb a sick employee by anything intrusive like a phone call. This is not an emergency, neither for the company nor for him.

Send him a message, maybe EMail or SMS if you have his cell number. I will shamelessly plagiarize the existing answer and modify it a bit:

Hello Eric. Sorry to hear you feel sick, I hope you get well soon. Tomorrow is the conference. If you cannot make it, we need to send a replacement and they need to know before they leave the office today at 16:30. So please contact us before 16:30 and tell us if you can make it tomorrow.

The point here is to set a clean limit. It's not arbitrary, it's a deadline for the business requirement that the trip has to be scheduled and the person needs to plan for the following day. But it leaves no room for interpretation what happens if the employee does not communicate (because if they are sick and maybe went to bed or took a lot of painkillers, not communicating would be the default).