Can a company buy its own shares to manipulate prices? Can it reduce its capital by buying shares? In other words, can a company become its own member?
There are some interesting subtleties in the questions:
Can a company buy its own shares to manipulate prices?
Yes it can buy its own shares, but there is no practical reason for it to do so just to manipulate prices. Buying pressure raises share prices, so a company buying a lot of its own shares might raise prices, but it would be of no benefit to the company. A company cannot profit from the sale or purchase of its own stock.
Can it reduce its capital by buying shares?
By definition, buying back shares does reduce capital, since net cash is reduced. The shares it purchases become "treasury shares" that are considered equity (not assets) on the financial statements.
In other words, can a company become its own member?
Not sure what you mean by "member", but Treasury Shares have no voting rights, pay no dividends, and are not included in "shares outstanding", so the company cannot be a voting shareholder.
It sure can.
Nearly all large companies have at one time or another share 'buy-back' programs, where they buy their own shares back. Those shares are normally just considered non-existant for any voting puposes, so there is no problem with circularity.
As a result, the share price might increase (often the reason for doing it), but the number of reported 'outstanding' shares shrinks, so the total 'value' of the company doesn't necessarily change (unless the other shareholders consider it a value increasing move, but that's their decision).
Decisions for buy-backs are often made by the shareholder meetings themselves, so shareholders are not 'surprised' by it, but that is not required.