And this is the way it's supposed to be: High earners will be net losers, people with low/sporadic/zero market income will be net gainers.
I don't know how strong the effect would be, but if true, the effect of basic income would be a strong subsidy from taxpayers to real estate companies.
This will never happen (through government legislation). If it does happen, then the policy will encourage individuals not to work, since the income to work ratio is better if you make less money. This would be a disaster for a country's economic development.
In reality, there is already a large basic income program, it's called Social Security.
I'm sure you probably think that giving millions of elderly the freedom to retire is terrible for economic development, but it is no doubt a great advance for human decency.
But who says income tax is the only type of tax? You can use a progressive consumption tax (no tax under $x item, y% tax from $x to $z, etc) to raise the funds. Economists much prefer this type of tax because it disincentives consumption rather than production.
The rent will precisely increase by 1000 Euro. Yes you can compare percentages all you want, but at the end of the day, 1000 Euro basic income is 1000 Euro more for everyone. That will be directly plowed into everyone's (for example) personal rent/housing budget or tolerance, which translates to inflated rental prices.
If I make $15K/month and receive a $1K/month basic income, I'll gladly pay $1K/month more for rent -- just like the guy making $5K/month would. Can you tell me what will happen to rent rates?