If you think of a person as having a balance sheet, with assets (property, shares, bonds, cash) and liabilities (loans), and a profit and loss of revenues (salary, dividends, coupons, rents, i.e cashflows) and expenses (food, shelter, clothing, transport, interest etc), then inflation:
* increases the value of your assets => asset prices rise with inflation
* decreases your liabilities => loan balances stay constant in nominal terms, but in real terms the liability is decreasing.
* increases your income => wages and salaries rise with inflation
* increases your expenses => the price of consumables increases with inflation.
So, ideally, in a high inflation environment, you want to hold as many assets as you can afford, levered as much as possible, with a high income, and low expenses.
So the rich purchase assets with debt. Inflation pushes the price of assets up, and the real value of the debt down. I agree that the assets you purchase should generate a cashflow to cover the financing drag. Inflation also increases dividends, rents and coupons.
Because the poor have expenses (outgoings) equal to or greater than income (wages, salary), price inflation erodes their disposable income, and price inflation is elastic, but wages are sticky, so the poor are always playing catchup, and in the interim, the rich are buying up their assets with cheap debt.
The rich are both lending and borrowing money; again, who do you think they are borrowing money from? Who are they buying the assets from? Again, it is not the poor, unless you are suggesting they are accumulating previously non-existing or non-utilized assets?
The rich are not borrowing from each other. Banks create money endogenously. They are buying existing assets from each other, and they are also buying new assets as they are created. Without inflation, asset purchases would look less attractive on a cash return basis, so inflation does serve that purpose. However, persistent inflation is forcing the market to be investors when they may want to save or spend.
If you have savings and investments, inflation puts pressure on you to make them perform, otherwise you are losing money; it works like a capital tax in that regard. You just can't sit on it since money can't really be saved without someone else borrowing it (production and consumption have to even out at the end of the day!). So no, the rich don't really get richer off inflation. They aren't brorowing money from the poor at any rate.