I'm maybe not experienced enough as I only do small electronics projects with a limited set of components which I always use, however, in my limited experience I see typically see datasheets in two places: on the manufacturers website and on the distributors website. For example, if I buy an Microchip EEPROM from Mouser, then the datasheet is typically available both on Mouser's product page and on Microchips documentation page.
Very much so, especially for the types of parts you will find avaliable from the Chinese sites listed. Finding a data sheet for an AVR or a MSP won't be a problem, but finding a data sheet for a super-cheap off brand micro can be a real challenge.
For hobby electronics projects, the only time you may run into this is if you are using recycled components, since datasheets for older parts that aren't current stock may be very hard to find.
I guess this isn't always the case?