Depends on the company. Many of the national/international companies will have dedicated servers that they own, but there are plenty of business that rent dedicated servers from a hosting provider, and I can't blame them. Server farms get to be expensive to maintain.
As far as the number of pages, it is largely irrelevant. It's really the bandwidth that counts and 1000 games on one pages takes up just as much digital space as 1000 games on separate pages. To tell the truth, many of the sites that index a large amount of content often have few to no pages. All the "pages" can be compiled server-side as the client requests them.
Long-winded rants aside, go for a well indexed, easily navigable layout. It's worth the extra trouble up front.