Information is latin, not English. Therefore you find it all over the European languanges. The difficulty with the English version is that it is an uncountable noun while "informationen" are perfectly countable in German. Therefore you don't have a plural with s at the end in English which is a typical trap for German speakers, just like "I become a sandwich"...

Trainieren and Training are completely normal anglizisms in German, originally used only in a equitarian context. Today you can not only train your horses, but your staff as well...

I think a native speaker, even a yankee, would be more likely to say "...training course/seminar" as 'training' is just to vague, using it as an adjective rather than a noun especially in this context, though I'm old school and hang with the pedantic kind who also use complete words and sentences in texting so I may be biased.

However, just because it has the "-ing" suffix doesn't make it a verb necessarily. It can be both, depending on how it's used.

But would anyone use them together like that as a single something. ...training information? I can't think of a use that might apply and not sound odd.

If it was information learned in a training course, I think you'd be more apt to say course information rather than training information.

Training is plain English, but you cannot say "I am attending a training tomorrow" or "I went on three trainings last year"... although in reality you hear this kind of thing all the time in multilingual offices.