Aren't there any other alternatives for this acceptor?
Not that we're aware of. Every other alternative requires an anaerobic environment - which means small, and often less efficient.
Oxygen is already recognized to have several harmful effects to cells - wouldn't another molecule be a better choice?
When we're talking about a molecule's fit there are many things to consider. Primarily its electronegativity, as that determines its stability, ability to accept and donate electrons, protons, and its ease of acquisition given plentiful amounts in the environment.
Molecular Oxygen is comparatively stable to a lot of di-atomic molecules, and comparatively reactive to others. It's in the butter zone for the act of being the terminal acceptor.
Let's think about Fluoride: Almost all of it is bound up in rocks with other elements. It's not available everywhere. It might be an alright candidate if we could eat rocks, but it also replaces minerals in our bones and teeth. Yes, you're familiar with this - but too much leads to fluorosis - where too much of the calcium compound is replaced, and it ends up making the bones and teeth much weaker.
Alright, let's think about Nitrogen - Very stable, pretty good electronegativity, abundant... but a little too stable. Yup, it's a gas, it's everywhere, and we can't use a lick of it. It's so stable that it's basically inert. The only reason rhizomes and other beneficial bacteria can use it is because they're in a controlled environment away from oxygen with significantly different biochemical processes which produce much less energy (and probably wouldn't be able to support large multicellular organisms that are as active as we are). So, Nitrogen is out.
Carbon? Nope, as an ion it's dangerously reactive, and as anything else it's too stable. Plus, we're carbon-based! If we used carbon as the terminal acceptor then the proteins handling the electrons would have to be even more exotic and filled with other elements to prevent them accidentally gobbling up the electron and becoming useless.
Sulfur? Not electronegative enough for human purposes, unfortunately. It wouldn't act as a terminal acceptor unless our bodies seriously oxidized it; and any oxidized molecules floating around would be more dangerous than molecular oxygen floating around. Plus most sulfur is bound to minerals as well - not abundant unless you're in the ocean.
Oxygen is both incredibly abundant and easy to perform the chemistry with. More reactive molecules are usually not abundant and tied up in the Earth's crust and could be more dangerous.
Less reactive molecules tend to be so non-reactive that we can't do anything with them, and if we could force them into a state where they would be reactive they tend to be violently reactive or require very specific environments that we can't reproduce on a scale necessary to be as mobile as we are.
Oxygen is not perfect, but it's a good fit.
Why does it even require an "acceptor" to accept the electrons and hydrogen ions? What would happen if they were left alone?
Same thing that would happen if you clogged a pipe or hose. Once all of the electron acceptors on the transport chain are full, movement stops. The electrons won't be magically absorbed by other parts of the proteins making up the chains, and while there might be errant reactions with other molecules, you'd be lucky to get a whole ATP molecule out in a day - assuming you could magically stay alive.
In reality, Cyanide poisoning is precisely what you're asking, and will kill you very quickly.