What I don't understand with BI is this: if you give everyone the same amount of money, won't prices and rents adjust accordingly by exactly that amount, thereby nulling the effect ?
Yes. And then the BI payments will be CPI adjusted, engineering an inflationary spiral, or price controls will be introduced, engineering chronic commodity shortages.
1> Proposals for the institution of BI as a public policy do not create money out of thin air, they redistribute it from wealthier individuals to poorer ones. (So if you earn $2million per year, you might pay about 1million in taxes and then get back 10thousand in BI. If you don't work, you pay no taxes and get 10thousand in BI.)
2> This YCombinator experimental program is not a fully implemented BI system. They will be selecting low income individuals and paying them a regular no-strings-attached income to see how it affects their lives/behavior. Other people in Oakland who do not receive these payments will probably bear a cost due to inflation as more money ends up in Oakland.
I think that what you would see is rents exploding in certain places like SF, but if you don't need to be in a particular city for work or social services, a lot of people would move to places like Cleveland or Detroit that are much cheaper to live in.
Detroit is cheaper right now, but people aren't moving there. Post-BI, Detroit landlords will raise rent to capture any additional disposable income, so everywhere will still be in the same relative position.
Possibly (though many polities have some form of rent control). There's lots of opinions about what would or wouldn't happen. The difference is YC is going to test it empirically.