Recall the basic principle of copyright law, as detailed in 17 USC 106:
The copyright holder has the exclusive right to make copies; prepare derivative works; or distribute copies by sale, rental, lease, or lending. Other people can legally do these things only if they are given permission by the copyright holder, typically via a license. (Remember, the literal meaning of the word license is permission.) Often, the copyright holder will require a prospective licensee to accept various terms and conditions before the license will be granted.
If there is "no EULA", or if there is one but the purchaser has not agreed to its terms, then the purchaser has not been granted any such license, hence does not have permission to do any of the things listed above. If they do so anyway, it is illegal copyright infringement and they will be liable for damages.
To use a firewall analogy, copyright law is "default deny".
So let's take your questions one by one:
- Do they own the software?
US law has no concept of literally owning software. The closest thing is owning the copyright, which the purchaser certainly does not. It still belongs to the vendor that wrote the software (or whoever they may have later transferred it to).
- Can they legally alter the code of the program they purchased?
No, that would be preparing a derivative work. The copyright holder has not granted them a license to do that. (There are some exceptions for purposes such as reverse engineering and interoperability, see 17 USC 1201(f)).
- Can they legally redistribute it
No; again, that is the exclusive right of the copyright holder, and the purchaser has not received their permission.
Maybe, if the first sale doctrine applies. Its application to software is complicated. The user has a better case for being able to sell the software if it exists as some tangible object which is transferred (physical media, pre-installed on hardware, etc).
- Can they legally modify the code of the program for others who have also purchased the same package?
No, that would be preparing a derivative work.
- Can they take that software and install it on a secondary machine?
No, that would be making a copy.