Are notes played on the beat [always] played louder?

Coming from the perspective of tracker modules and composing in a DAW, there are advantages and disadvantages of each approach.

You could keep the volume of the main instrument the same (or decrease it slightly) when it comes in tandem with a drum beat. This keeps the overall track volume more regular, at the cost of the drum beat "competing with" the main instrument).

You could boost the main instrument but that would create awkward high-amplitude "pulses". If you use software tools to limit the maximum volume at any point, it will make the rest of the track relatively softer which may not be what you want. In some genres (especially electronica, EDM, and "club" or "rave" music), such high-amplitude pulses are considered a main feature rather than a drawback.

Ooh. You are opening a great big box of musical history. Just to make it simple, look into a style called "bebop". This migth be one link: one example . To make it very simple, bebop players changed that accents all the time using it to accent the melodic line.

If you want to make it very simple, accent the off-beat notes, and perhaps the 4:th beat in the bar most.

Well, they are accented by nature, but what it does mean, that's a different story. Consider the motif of Bach's small fuga G major (BWV 557) (source):

The 1st and 3rd beat of the beginning of the Fuga are empty, yet there is an accent on them (Bach was very systematic, so yes, there is). Also, organ doesn't really have "loudness" for single notes. So yes, they are accented, but how it transforms into the performance of the piece may be very tricky.

Last but not least, note that in modern music, you often have off-beat accents, but from the performance, it will in most situations still be clear where 1st and 3rd beats are.

"I know that the 4/4 time signature is played as [ONE-two-Three-four]."

How do you know this? Is this from standard music training?

In any time signature we need to somehow express the periodicity of the meter. There is a concept called metric accents. In reality you don't "need" to accent the strong beats of a time signature but without them the "feel" will start to fade and both the musician and listener will have a harder time following the music. Music is divided into short phrases and accenting helps to define these phases as much as melody. While it is true that in 4/4 beats 1 and 3 are generally given "metric accents" it isn't always necessary. Accenting beat 1 alone may be sufficient to define the phrase and keep time.

Accenting the down beat of three on the second group of eighth notes is not wrong, it will create one particular feel. But accenting on the up of 3, {Ba, ba-da, ba-Da, ba} will create another. When accents are missing it is up to the musician and their experience to introduce accents that are typical for the style of music, German, Latin, Afro, Metal, etc. So taking the single measure out of context and trying to put a hard and fast rule on it will fail. I would use judgment based on the phrase in context with the rest of the music. With no other context I'd accent only beat 1 and run through the eighth notes to beat 4 with no other accents.

However, I sometimes only accent the first or last not of an entire phrase even if it's 4 measures long (rather than accent 1 and 3 of every measure). And sometimes I accent the highest not of a run regardless of what fraction of a beat it lands on. Rising up in pitch and volume don't always have to correlate but when they do it creates a very uplifting feel.

For a beginner learning to tap a foot while playing there is nothing wrong with "accenting" the down of 1 and 3 all the way through. But artistically, you are free to do whatever seems right. Don't accent at all is an option.

If you insist on accenting beat 3 I'd accent the pair STRONG-weak.

Good question! Made me think. The first beat of a bar - any bar - is generally accented, but by how much varies considerably, and sometimes depending on the phrasing, and the actual notes involved, it's not accented at all.

Had there been a phrase mark over all six notes, then I wouldn't expect any accent on that F note. As it stands, it could have an accent - but very, very slight.

That's one of the myriad of decisions that are made by the player, and it may well be different next time. Often, it's not even a planned decision, it just happens. So, although the question needs asking, there won't be a concrete answer!

Interesting thought - if that F was a 'pushed' note - a semiquaver earier, so there was no actual note hit on beat 3, (it's tied to another F semi on beat 3) then more than likely that would be accented. Just a thought...