‘What are people’s thoughts on using the office gym during the day?’

It’s hard to say without knowing the culture of your firm.
Generally, when you are a trainee or a pupil it’s usually better to take the path of least resistance and avoid doing anything that people can take objection to. Final recruitment can be quite political & you don’t want triviality to be used as evidence that you aren’t committed , skiving off during the working day when others are working.

Lunchtime is probably acceptable but outside that I’d be careful. If you are outstandingly brilliant it probably won’t matter but if like most people you are normal it could be used as a hook for negative feedback.

It’s only 24 months. It’s not worth it.

I’d say it’s possible if you’ve built up some reputation as a reliable pair of hands in the first few weeks/months of your seat.
Listen to music on your work phone and keep off silent mode. That way you’ll know every time you get an email or message from someone, and if you need to respond then you can.
And if someone asks about your whereabouts and needs you back at your desk, then make sure you’re able to be back in 5 minutes. So don’t work out so intensely you’re sweating everywhere.

Obviously don’t do this if the team is working towards a deadline, only for quiet moments (as already stated).

Fireable offence.

Invite your supervisors to the gym. If you can lift more than your supervisors, you automatically become their superior