The USPTO certainly seems to think an ISA is patentable, and I haven't seen a court disagree yet.
[1] https://en.wikipedia.org/wiki/Alice_Corp._v._CLS_Bank_Intern...
Yes, any software patent is possible by describing it as running on a processor. See for example [1], which has the very common claim prefix of "A machine readable storage medium storing a computer program..." Alice doesn't invalidate these patents unless, by removing that text, the remainder of the claim is unpatentable.
Because software patents are still legal, there's no need to attempt to describe them as running on a custom hardware device - you just specify them as software. Specifying custom hardware would unnecessarily reduce the scope of your claim.