For proprietary software you could have licenses. In fact, the reason that IBM first licensed some of its software for the IBM/360 was that it was unclear at the time whether software could be copyrighted.
However, absent copyright, it would probably be impossible (IANAL but have talked to lawyers about these topics) to enforce any usage provisions on open source software where you haven't explicitly agreed to a license. (Open source licenses basically give you rights that you wouldn't otherwise have under copyright law.)
See above comment: the question is not "what does copyright law say now?" but "is some variation of that appropriate as a replacement for patents?"
In other words, all options are on the table. We're assuming that Congress does its job, for once, and being that they're politicians, they do sometimes respond to public pressure.
Well, it's not just Congress. There's the Berne Convention which, with minor variances, governs copyright in most of the world. The US can do whatever it wants I suppose but it's not as simple as Congress saying: We're going to do our own thing. Who cares about Europe etc.?
I am not a copyright law expert. But the Berne Convention originally dates to 1886.
There is apparently some flexibility within the Berne Convention on copyright terms. There are also limitations on public domain works in continental Europe that differ from common law countries like the US. However, it does govern copyright in broad strokes. For example, the US used to require that a creator explicitly assert copyright while most of the rest of the world did not.
In any case, the US is generally aligned with copyright in most of the rest of the world. So any broad change in copyright law (other than perhaps somewhat shortening terms) would make it an outlier.
However, absent copyright, it would probably be impossible (IANAL but have talked to lawyers about these topics) to enforce any usage provisions on open source software where you haven't explicitly agreed to a license. (Open source licenses basically give you rights that you wouldn't otherwise have under copyright law.)