The Sign Language Used

I think it's different than standard ASL as Helen Keller was not just deaf she was also blind so the sign language would also have to be haptic so that she could "feel" the words. The alphabet signing looked to be the same as ASL though. Maybe an expert can chime in.

NEVER KNOCK ON DEATH'S DOOR.
Ring the doorbell then run.
He totally hates that.

What Helen and Annie are doing, throughout the film, is fingerspelling -- spelling out each word. The hand positions they are using for each letter look the same as the modern ones, to me.

However, it's not what most people who sign are using these days. In American Sign Language (ASL), there are single signs (hand positions and/or movements, often matched with facial expressions) for whole words, and even concepts. Only the things that have no single sign (often proper nouns) get spelled out.

For the sentence "I'll visit my mother in Altoona" the only thing that would be spelled out letter by letter is "Altoona."

So, Annie and Helen would have been using a single gesture each for:

doll - http://www.lifeprint.com/asl101/pages-signs/d/doll.htm
tree - http://www.lifeprint.com/asl101/pages-signs/t/tree.htm
teacher - http://www.lifeprint.com/asl101/pages-signs/t/teacher.htm

etc.

Grammar and word order are often different from spoken English. And, as with any language use, there are puns, "accents," slang expressions, professional jargon shortcuts, nicknames, and disagreements about what is correct!

I imagine things differ in other countries - I just know the English alphabet and a bit of ASL.

If you want to see signing that is more like contemporary use (I think it's essentially ASL, with fingerspelling only when needed) in an older film, watch Johnny Belinda. And, in just about any newer American film in which people sign, you'll be likely to see much more ASL and very little fingerspelling.