You are only legally compelled by the government to fill out tax paperwork if you experience a taxable event in that year which would require you to fill out the form. If you do not have a taxable event (i.e., no income) you are not required to send any information to the government in relation to taxes.

https://en.wikipedia.org/wiki/Compelled_speech

EDIT: IANAL, please do not cite this as any legal justification in a court if you fail to file any form with a government.

The point is that the government can compel speech without endangering the first amendment. The IRS could decide tomorrow to mandate a form for those below the taxable level without running afoul of compulsion.

If your income is subject to taxation, you can't refuse to fill out the paperwork on the grounds that doing so lends legitimacy to an institution you don't support. The same should be the case for voting.

What would be the requirement to compel voting? Be of voting age population and be mentally sound? Sounds like a massive expansion of state power to force all people of a certain age and sound mind to do an action.

I'm not sure what you mean by the "requirement." In order to vote in the US, you have to be of voting age. Some states impose additional requirements, like not being a felon (which I think are wrong, but that's another story).

Mandatory voting would have the same requirements. Anybody who previously could vote now must.

In terms of state power: it's not all that great of an expansion. If you make over the taxable limit, you pay federal taxes. If you own any sort of land, there's a good chance you pay local taxes on it. These sorts of compulsions apply to (equally?) large swathes of the population without necessitating a great deal of state power.

There is a massive difference between paying taxes when you choose to participate in an economy than having the government compel you to make a political statement. You choose to buy land which is protected by the state. You choose to buy things at the store. You choose to have an income. You can choose to not buy a house, you can choose to not work for an income.

The closest thing to compare compelling people to vote would be the draft.

You think people are less compelled to have an income than they are to vote if a small fine is levied? It doesn't make any sense. You put in a small fine for not voting, you make it clear that you can walk into a ballotbox and draw a huge x and stick it in the machine and you don't have to pay the fine and bada bing bada boom, we've got a lot more voters.

That's a strange sort of "choice": if I don't buy things at the store, I starve to death. If I don't buy land (or rant it, or inherit it), I die from exposure.

Like I said above, a mandatory vote does not require individuals to make a political statement any more than taxation does, unless it is coupled with a limited selection of candidates. We can talk about the propriety of a mandatory vote, but the legality of one in the United States seems fairly cut-and-dry.

Do people who experience taxable events (possibly without consciously choosing to do so, e.g., inheriting money or property) lose their First Amendment rights? How?

If you chose to not take the inheritance, you would not experience the taxable event. You are not compelled to take an inheritance, the probate courts would absolutely love to take the propery if you do not want it.

It is very different. Compelling one to fill out a ballot (even if it is blank) decays the right to vote into the obligation to vote. Paying taxes is an obligation. Choosing whether or not to vote is a right.

Not to mention Selective Service (as in, the form you must fill out).