In addition to what KFdancer has already said.
If the visa does not have long to expire, the UK do not need to curtail it as it will just expire. To avoid becoming an overstayer, he would need to leave the UK or put in a
valid application for another visa before his visa expires.
Is it possible to get sponsored by your employer in this instance if you have a particular job or meet a particular income threshold?
It would be under the same terms as anybody else who wants to come to the UK on a Tier 2 General visa. He would also need to apply for this visa from the US.
https://www.gov.uk/tier-2-generalHowever as he has been a student in the UK on a Tier 4 visa, he
might be able to be exempt from the RLMT. Once he is on a partner visa, he won't be exempt from the Tier 2 General RLMT anymore.
As he wants to live and work in the UK, was he unable to find a work sponsor while he was on his Tier 4 visa? If he could have, then he would have been able to switch (in country) to a Tier 2 General visa from his Tier 4 visa.
Is it possible to meet the extension and ILR requirements even if a couple have broken up?
His 5 years to ILR clock would stop. It would start at zero again on another visa route that leads to settlement (ILR).