First Question Ans:
By right after, I really meant at least a week or two after the National Holiday. Oct 15 - 30 would likely be less crowded, but that was about the time in October (2 weeks after the National Day holiday) when my wife and I went and she said afterward: "glad to have done it once but never again" because of the crowds. During the National Day Holiday week though, many people (hundreds) who wanted to get on a bus to go down to the entrance at closing time couldn't get on one until around 10pm and it was not warm there. There were near riots in the park. So mid-October is much better than during the holiday. Also plan on going mid-week day(s), not on a weekend.
The largest crowd is at the lower level just past the park's entrance where you get on the bus to travel up to the various site stops. When we got there, I'd say there were several hundred, maybe even up to a thousand folks in line in front of us and it took the better part of an hour from getting our tickets to boarding a bus. Actually this part was pretty well organized, with several buses boarding at one time (maybe 8-10 at a time) and as soon as a bus left it was soon replaced with an empty one. If you can, get there right at opening time when it should be least crowded.
Most buses make stops traveling up through the Shuzheng Valley to the tourist center located at a Y intersection where you then get on another bus to continue farther up one of the sections of the road which diverges at the tourist center to the Zechawwa Valley (left leg of the Y) or to the Rize Valley areas.
At the Visitor Center and at many of the stops, getting on the bus was not easy. Had to push and shove like the Chinese to get on. Fortuately I'm a big guy so my wife would just follow behind me in my wake. Some stops no crowds and easy to get back on the bus while at others we had to wait for 3 or 4 buses to pick up passengers in front of us before getting on. If you are shy and polite about making your way forward to get through the bus door, you'll never get on a bus at some of the stops.
We were there two full days, first on the first day making a stop while going up the Shuzheng Valley and then busing to the last stop in the Zechawwa Valley and treking down. The second day we did the Rize Valley side and lucked out in that it was so busy at the tourist center, the bus just continued on up the right Y of the road going to the Rize Vally, where we were planning to go, saving us a good hour waiting to board a bus. If I only had one day, I'd go up the Rize Valley first and then see what else I could fit in. Two full days are highly recommended, but the entrance ticket is a little pricey.
Once off a bus at most stops you could usually find a fair amount of breathing room and the scenery is outstanding. If you go all the way up on either of the roads, get off at the last stop and then trek down, so you can minimize the number of times you have to get on a bus, but eventually you will have to get back on one. If nothing else, it's an experience you'll likely remember for a lifetime.
Answer to your second question: I don't know what a car and driver will cost you - it varies on several factors, but I can give you a general idea.
In Guilin, you would mostly get around by taxi or bus; or boat to Yangshuo, though going direct by bus is an option too. Taxi between the airport and the Waterfall hotel area in Guilin would be about 125RMB or take a bus for maybe 30RMB/person. For the various attractions, you can of course join a bus tour; for example to the rice terraces or to Yangshuo (or by boat then a tour bus), but again prices will vary. A rough guess if you decide on a car and driver for a day (6-8 hours) would be around 600-800RMB/day, but this is a swag.
In Jiuzhaigou my wife arranged for a car/driver to pick us up at the airport, take us direct to Haunglong, wait for us while we took the cable car up the mountain and then trek down for about 4 or 5 hours, then drive us to our hotel near the Jiuzhaigou Park entrance. The next two mornings/afternoons took taxi to the park entrance and back for the hotel. The morning of the 4th day, the driver from the first day picked us up and took us back to the airport. I seem to remember that the airport to Haunglong to Jiuzhaigou, then Jiuzhaigou to the airport total cost was around 700 to 800 RMB , but it's about 3 years ago, so might be higher now. I understand that there is a bus from the airport that can be taken to Haunglong and back to Jiuzhaigou, but don't know the cost or the schedule.