Before the 3rd century you had to be a citizen of Rome to join the army. In order to acquire citizenship if not born to it back then , you need to speak Latin and learn about Roman culture. However during the 3rd century the qualifications for citizenship were lowered , and into the 4th century legions were made entirely made of foreign mercenaries with no attachments to Roman culture and language. The commanders here were also foreign and spoke the native language of the men.

Tldr: they originally did then the rules got relaxed

It's worth keeping in mind that even as late as the 2nd century AD, many of the inhabitants of Italy didn't speak "Latin", but continued to speak other Italic languages, like Unbrian, Oscan, etc.

Because of the nature of spoken versus written language, our knowledge of Vulgar Latin is limited. However, I think you should make an effort to re-examine your own understandings of what Latin actually was. Even in Rome, 'Latin' as it was written in the classics was not being spoken. Outside of Latium, there were a wide variety of Italic languages being spoken. Even in the middle Empire when Rome was at the height of its power, Latin as we know it was more of a written than spoken language. So, that said...

By the third and fourth centuries A.D., were most of the army's recruits not native Latin speakers, speaking mostly other languages or a dialect of Latin?

Define 'native' Latin speaker? If one is speaking a Gaulish vulgate of Latin, is one not a native Latin speaker? It's difficult to qualify.

It's my understanding that orders were given in Latin, so did soldiers have to learn how to speak it? If so, was there any formal instruction?

I don't know of any formal instruction for language in legionary training. That said, it is likely most recruits to the legions would have had enough commonality in the dialect they spoke to make basic orders intelligible. The specific variety of Latin they ended up speaking in the camp is another story though.

Was Latin the "lingua franca" of military life, with Latin the only way men recruited from different linguistic regions could speak to each other?

As twihard mentioned, prior to the Edict of Caracalla in 212, Roman citizenship was a limited status, and only citizens could be Legionaires. It would be a safe bet that citizens and their families, who made up the Romanized populations accross the Empire, whether in Italy or in Syria, would have been able to speak at least some manner of Latin. Auxilaries are another matter.

Could this rough "army Latin" have helped to create the perception that the army had become "barbarized"?

I think that this is just an example of the fickleness of primary sources. Dio was not particularly fond of Severus, and Severus himself did not have the best relationship with the senate, of which Dio represents the viewpoint of.