The homeless vs middle class inequality could easily be much the same as it always has been. The middle class are getting increasingly tenuous in their wealth.
I expect that when most people talk about the gap they are referring specifically to the actual conditions of the poor, and not the range of the gap.
Inequality problems can be very effectively by taking stuff away from the wealthy. That doesn't work for a living standards problem because there isn't enough real stuff for it to work. Look at the real amount of resources consumed by the mega wealthy and divide it by the number of poor people - the results aren't very impressive. Not the paper wealth, but real consumption.
Jeff Bezos for example may make billions on paper, but his actual personal consumption isn't going to be anywhere near that amount in goods and services. Distribute his real consumption to the poor in New York and it won't make much of a difference, especially over the long term. But absolute inequality would noticeably drop. The poor would appreciate it, but it isn't going to solve systemic living standards problems.
[1] https://www.nytimes.com/2002/02/13/world/sao-paulo-becomes-t...
I don't understand your reasoning. If some people are homeless while others have houses, how is that not inequality?
I've lived in São Paulo, where homeless people can be found on literally every street, while millionaires fly from skyscraper to skyscraper in their private helicopters to avoid the impassable traffic. No-one needs to pore over dry statistics to know that Brazil (to take just one example) has enormous levels of inequality.