Interesting armor and discussion.
Maybe I should post a picture of my own armor, it looks a lot like the white laced armor in the first picture, but with gold flanges on the helmet and with dark blue lace.
Why no picture of the very rare horse barding?

The rope is the chin strap. Overstated so it could be tied into a big, gaudy bow knot or something, as only the Japanese could do. They always seemed to make their chin straps out of half-inch rope or larger.
The "shinobi-no-o" is about as thick as a finger, and yes it was typically tied into a big gaudy bow

The hooks are to hold the front crest. From the size of them and their position, I'm guessing that the helmet had horns originally. (Deer, cow, who knows?) Probably big ones. Although it could have been some other preposterous front crest - flower arrangement or a stuffed duck or something. Whatever it was, the hooks held the crest. The helmet is tight-pattern lace, so it was expensive ... so I'm betting the original crest for it was totally outlandish.
I have never seen a stuffed duck on a Japanese helmet before, but I kinda like the idea

Also, note that the more outlandish helmet structures can be fake, they were made about 50 years ago, and most crests seen on antique armors are modern and mass produced... or occasionally a somewhat well made fake.
Even famous museums have and often still display reproductions or fakes.
But you sure are right about the very large horns, it would have had a large V shaped stylized pair of horns, they were common to the Ii clan, who were pretty much the only samurai with red armor during the Edo period, so it's very easy to attribute the helmet to the Ii clan.
The red helmet does look like it's in very good condition for it's age, it is possible that it has been partially restored, but I would need to take a close look at it, the mabezashi (visor) does look like it has some wear spots on the gold lacquer, but this is difficult to see because of the low quality of the pictures.
This type of helmet is actually quite cheap though, as the helmet bowl consists of only three plates, and although full lace would have been more expensive, it would have been negligible in this case (it's not a cuirass with lots of lace after all).
Also, the simple flat plates of the neck guard would have been cheaper then plates with build up lacquer (kiritsuke-zane, made to look like real scales), and it would have been a lot more cheaper then real scales.
It sure is a nice looking helmet, and this type was very popular with both poor samurai as well as wealthy daimyo.
Examples of the horns on Ii clan armor:


And yeah, seeing these things gets most people interested. That's what drives historians, in a field that most people find incredibly boring when they read about it on paper ... but when you're actually holding a 400 year old helmet, or standing inside a 1000 year old tomb, just about everybody gets a lot more interested.
Indeed, there is nothing like getting your hands on antique weapons and armor, I have been very fortunate to have hands-on experience with both Japanese and European antique armor and weapons.