Steve, welcome to Flyertalk.

When you are logged into Flyertalk, you can use the search function. You can do a basic search or an advance search. If you just type in "Beijing," you will see that there are a lot of posts on travel to China -- many related to the questions you have asked such as hotels etc.

For some quick answers to your question, yes, you can rent a car and driver, but usually the only need for a car and driver in Beijing is for a visit to the Great Wall (either Mutianyu or Badaling). Your hotel can help arrange this, but usually it is around US$50 or less (but I'm sure others will correct if I am wrong), but your driver will not speak any English in all likelihood. The alternative is to hire a English speaking tourist guide as well. Not sure on the cost but probably still cheaper than paying for two people to cram into a minibus for a Grayline-type tour.

While the majority of Chinese do not speak any English, it's probably a safe bet that you can get by speaking English at major tourist sites. For sightseeing in Beijing proper, you really don't need to hire a driver. You do need to have the name of your destination written in Chinese characters though to show the driver. Your hotel can assist you on this.

Five days in Beijing is almost enough to see the main sites. October is a GREAT time to visit -- weather isn't bad at all, actually quite nice.

After Beijing, the three other most common destinations are
Shanghai
Xian
Guilin

More adventurous souls will venture to
Lhasa
Chengdu (panda reserves)
Yunnan province (ethnic minorities)

And even more off the beaten rack areas would include
Inner Mongolia
Silk Road towns (Dunhuang, Turpan, Urumqi, Kashgar)

Where else you visit really depends on your interests, time, and budget...