I originally came to know bitcoin in 2011, did some basic mining, and eventually deemed it not worth my time and thought of it as a sort of internet fad. I now realize I was really just sowing the seeds of regret. Anyway, I did some more research into the technical aspects (though I didn't read the author's full papers) and was hoping someone could clear up some things.

For one, if transactions are processed and confirmed in new blocks that are mined, then what happens when new blocks become scare and are mined very slowly at the rate of say 1 block per day for the whole network? Bitcoins are finite so doesn't that mean that new blocks are finite and thus new transactions will be slowed to a craw? I feel like I'm mistaken here, so what will really happen to the way miners process transactions when bitcoins/new blocks become scarce?

Secondly, if every node has to download the entire block chain which is a history of all transactions, won't the block chain quickly become extremely large and force normal PC's out of the network due to hardware requirements?

Thanks for any info.