What can Congresscritters expense? I'd assume they need to travel between their home state and DC frequently. Would lodging for work where they do not have a home be expensable somewhere?

Isn't the solution to needing two homes to do the job for the government to provide that as a work expense rather than permitting funky financial hijinks?

It’s better than it used to be. Since 2023 House members can expense up to $30k / year in travel costs and lodging in DC. Prior to that, they had to eat those costs which were estimated to average around $28k per year based on number of days in session.