Are we stuck in a moment with the river of past and future flowing over us, or are we flowing down with the river of time into the future?

The ‘river of time’ is actually the river of change (the only real evidence of Time flowing is change). Time (in this context) is merely a collective noun.

And we all change…continually. So we change, with everything around us changing.

But we all change independently. Change is reference frame specific.

So the big collective flow (Time) and the indivudual events (from quarks decaying, atoms vibrating, life breathing, earth spinning, universe expanding…depending on reference frame) have their own specific ‘flow’.

https://qr.ae/pGByob

If the past, present, and future are entangled, are they happening simultaneously?

No.

The past changes to become the present which changes again to become the future.

This relationship is inherent in the core meaning of these words. So when people hypothesis about a different relationship, i’d say you need to go back and consider what past (has been) present (is) and future (will be) mean.

Not to forget that knowing the past relies on perfect memory or evidence, and knowing the future relies on perfect hypothesis and pattern understanding.

https://qr.ae/pGByHS