What is your Intelligence score?

I think we can look at this like a program. We are doing so ONLY as a method of thinking about this problem, NOT because this is how the game works. Below I will describe an update loop and the like, but the game does not run on such a thing. This is based on the phrasing.

The headband is not a one time occurrence, it is a continuous loop. It uses the word while.

Your Intelligence score is 19 while you wear this headband.

If we look at it like a program, we're setting the Intelligence to be 19 every "update". Or rather, every moment in time. Is the headband on? if yes, your intelligence is 19.

Feeblemind is a one time effect, meaning one time and one time only, we're setting the creatures Intelligence and Charisma to be 1. It has a duration of "Instantaneous", which implies that the effect is not contingent on the duration of the spell -- It is not 1 "while this spell is active", like the way the headband sets your intelligence to 19 "while you wear this headband". It is just set to 1. Without interference, your intelligence will stay 1.

When the headband is considered, your intelligence becomes 19. You can wear the headband because that is not the same as activating it, which the spell prohibits. If the headband's wording used "when" or "after" you put the headband on, we could conclude that Feeblemind would win out, as it was the latest effect on your intelligence. But because it uses "while", we treat it as a continuous effect.

Can I communicate, understand language, cast spells, and activate magic items?

Because of the wording of the spell, you still cannot speak, cast spells, or anything like that. The spell does not give a condition on the secondary effects' end, other than the spells that can dispell it, so we must conclude that the effects are ongoing as part of the spell itself, not because of the mental scores of the target. It is not because of the intelligence score that you cannot do those things, they are separate from each other.

Furthermore, Intelligence is not described as being important for language. Page 12 of the PHB states that Intelligence is a measure of mental acuity, information recall, and analytical skill. On page 17 of the PHB, you know certain languages "by virtue of your race". You also acquire some from your background. In the DMG page 278 for the language of a created monster, there are no ability score requirements for language. There are some spells that don't work on creatures of poor Intelligence, some of them regarding communication, but other than that there is no other requirement to be able to understand or speak a language. If you want more reasoning, there is a very good answer for a similar question I posed regarding Intelligence and Language

We must treat the inability to understand languages, activate magic items, etc etc, as an effect that the spell is having on your character, not as an effect the low intelligence is having on him/her. Call it "brain fart on steroids" if you want. You're smart, but you just can't do certain things. Your Intelligence is 19, you're very good at recalling information and can make good saving throws, but you cannot cast spells, activate magic items, understand language, or communicate in any intelligible way.

Does a feebleminded creature lose proficiency in weapons?

There is nothing in the spell that specifies you lose proficiencies with weapons. There is no requirement on proficiencies that say you must have a minimum intelligence or charisma score.
Since the 5e adage is that "things do what they say they do", it is not RAW that a creature under the effects of feeblemind lose proficiency. If you're thinking that proficiency in a tool or weapon is memory, I would respond that it is muscle memory.

How does a feebleminded creature decide what to attack?

As the spell specifies "The creature can, however, identify its friends, follow them, and even protect them." (PHB 239), it seems like identifying friend and foe is innate. If you can protect a friend, you need to know what threatens a friend.
The last source of an attack on you or a friend seems like a clear threat.
The healer of that threat might be less obvious and would take some intelligence to deduct that they are a defacto threat.

Can a feebleminded creature use character abilities?

Intelligence and Charisma based abilities would seem out the window. If not specified in RAW, being unable to communicate makes persuading someone seem unlikely.
Physical abilities, again, appear innate. Having low intelligence doesn't stop you from being strong or having quick reflexes.

Supporting Examples

Crab.
Int 1, Dex 11, Stealth +2.

So a creature with 1 INT can be somewhat proficient in stealth.

Giant Shark.
Int 1, Wis 10, Perception +3.

So a creature with 1 INT can be somewhat proficient in perception.