1. Why are miners rewarded but full nodes are not?
It is possible to determine which blocks exist in the blockchain (the work that the miner completes and gets paid for). It is not possible to determine who is actually running a full node. Who would pay those running a node? How would they get paid? How would you prevent fraud?
Note that solo miners and mining pools generally need to run a full node. If they don't, then they open themselves up to attack from competitors. Therefore, as long as we have plenty of miners and pools, we should have plenty of nodes. Note also that the intended design of bitcoin when it was created was to have a relatively small number of full nodes with most users just running lightweight clients. We can certainly change our minds about whether or not we want that, but we can't easily escape the fact that it was designed with such a purpose in mind.
With all this talk about increasing the block size, won't the chain quickly become HUGE?
Firstly, the talk is about increasing the block size LIMIT, not about increasing the block size. It is quite likely that a larger limit will eventually mean that miners will create larger blocks, but if there isn't enough transaction volume to fill the larger limit, then the blocks very well may remain smaller for a long time.
Secondly, the word "HUGE" is very subjective. How fast are you predicting that the blocks will actually grow?
Wouldn't the network be stronger if nodes were rewarded like miners for storing the chain?
Perhaps. If it were possible to reliable do so. Have you come up with a solution for any potential sibyl attacks?
the more full nodes running the more robust and secure the network is, correct?
Having full nodes certainly helps to make it more difficult for any small group to change any of the protocol rules to suit their desires, but the network would still run properly if it was mostly just miners and pools that were running full nodes.
2. Can average people no longer mine?
Anyone can mine. But it requires a lot of effort and access to inexpensive resources to mine
profitably. If you want to mine at a loss as a hobby, it is certainly possible to do so.
Why is there not a cap on mining?
Because it would be impossible to enforce.
Can't you make a cap like 1% for a pool and .01% for an individual?
No. Since it is impossible to tell who is running a pool, and who it mining individually, it is impossible to enforce any caps like that.
3. Earning bitcoins seems like the most fun part of this whole thing.
So then get a job that pays in bitcoins, or convince your employer to pay you in bitcoins, or make use of bitwage.co, or start your own bitcoin based business.
When I first learned about Bitcoin I was really excited about mining. Then I quickly realized it was out of reach for me.
Yes. Mining is a competitive business. Only those with access to the inexpensive enough resources can do it profitably.
It seems like you'd have to buy a new rig every 6 months to keep up with the hashing power.
Probably.
Even then you have to join a pool in order to have a shot at earning anything.
That depends on how big of a mining operation you can afford to set up, how inexpensive the resources are for you, and how much volatility in revenue you can handle.
Wouldn't this network run much smoother and be more fun if anyone could mine?
Anyone can mine, but not everyone can do it profitably. Mining is a competitive business. Those that have the largest margin between their costs and their revenues have the best chance to succeed.