I'm not a bus expert, but I have some insight,more so in rail. I'll try to answer your questions within my ability. I some cases I have been privy to related discussions, so I know a little.
1. Why do buses cost so much? Is $500k each a fair price or are prices inflated by the federal government and corporate monopolies?
I know of no "monopolies". Sure transit companies are natural monopolies, but they get nothing by paying more for a bus. My impression is that the purchases are highly competitive.
2. What is the maximum fuel efficiency for a bus? What are the most effective technologies?
Max fuel efficieny depends upon how you calculate it. Airlines get about 90mpg, because they divide the planes usage per person. So if I take my Ford wagon on a trip with 5 people and get 20mpg I'm getting 100mpg as the airlines calculate it.
A bus on a heavy route is carrying maybe 40 or 50 folks so they get great mileage. That exact same bus on the same route a 0 dark 30 is getting terrible mileage, because there are 0 to 2 riders. That is a huge dilemma for transit companies, how do you match the capacity to the need?
One simple way is to encourage employers to stagger working hours. Helps transit, helps the highways too. It is tried off and on, but never really successful. But to rag on this a bit, who would run a factory one shift a day? Yet that is what we do with our office buildings, one shift a day. Huge waste of space and heat and infrastructure.
3. Why do we have bus routes? Why not require riders to subscribe to the bus and route only where there are sufficient subscribers?
We have routes because you need to know where the bus goes to get on it. Subscriptions mean you would have to sign up and then actually use it when the service starts. People don't work like that.
If you are trying to get commuters onto a bus or rail vehicle you need about 3 years to reach target ridership. Why? Because people buy a car, and then have it for about 5 years. Once the decision to purchase has been made they stick with it until time comes to buy another car. Then they may reconsider. The converse is true also, if you disrupt a service, and force people into cars, then it takes a long time to get them back.
Bottom line is that you are really trying to change long term behaviour patterns, not easy, takes consistency and time. Get someone to sell his car, then change or cut back his route, go from 30 minute headway to an hour headway, and you will NEVER get him back.
4. Why are interconnections so difficult and time consuming? Is it really that hard to match points/times on different routes? Why are connecting buses not associated so they coordinate?
This is a huge deal in transit, which is simpler than buses because it's easier to make the trains run on time. Tons of planning goes into trying to make the most logical "meets" so that you can run the vehicle efficiently on a simple schedule, and still have your operators make their runs.
Believe it or not, transit system DESIGN and starts with figuring out the runs and meets. They use computer simulation software to run "string charts". You even figure in the engineers potty break time at the terminal. This is done with the civil design, because you need to know you speed profiles, slow curve, station dwell times, etc. only then can you start on the signaling design to support this kind of operation.
On busses there are simply far to many dynamic parameters to do a solid design, you fiupgure it out and take your best shot.
There are specialty "run cutting" scheduling programs that seek to optimize these problems. The programs try to optimize driver availability to the route, so you don't end up with a bus without a driver. Bus drivers often have to work "split shifts" say from 6 to 10 and then 3 to 7. Great job eh? To account for drivers, or engineers, that don't show up agencies have an "extra board", folks that sit around waiting to be assigned a route because someone didn't show up. Of course, they need to know the route, or the engineer needs to be qualified on the territory, or they are likely to make a real muck of it.
Yeah, it really is that tough.
And that's not to mention all the interference they get from the public and pols. A particular Civi group wants a special service or stop, some constituent wants a stop near his church or store. Road work, traffic jams, hills in snow all frig things up.
5. Why does public transportation have to be egalitarian? Why not have first class seats and/or routes?
Really? Well it does. Air lines, Accela, Amtrak regional, commuter rail, subways, trollyies, buses, walk. That's the pecking order. And if you don't like that buy a car, hire a limo, or get a taxi.
6. Why is a rideshare concept not part of the public transportation function?
It is, for the handicapped. And just ask someone who relies upon those services to see how well they work. Dismal was an opptomist. My Wife has had patients with a weekly scheduled appointment. Same time, same day, every week. Many times they were late or she would find them in the waiting room hours after the appointment, waiting for the ride back. Just sitting there in quiet bone crushing desperation.
..........
There is much wrong with public transit in the US. I've made my career in and around it in various forms. I'm not a big fan. It can bring out the very worst in people.
On the other hand, they have a pretty damn tough job that, I. Their own way, moast try to struggle through believing in their hearts they are doing good for society and culture and their community. They are just average folks doing an average job.
It's about like congress.