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?
No. Blocks are, and always will be, mined at a rate of 6 per hour (on average). The mining difficulty is adjusted every two weeks to ensure this is the case even with increases (or decreases) in mining power. The number of bitcoins per block is what decreases to ensure the money supply is finite, not the number of blocks. Eventually, the number of new bitcoins per block will reduce to zero, and no more bitcoins will ever be produced, however blocks will still be mined because miners also receive transaction fees. Currently transaction fees are negligible, however that will change as transaction volume increases.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?
Every full node has to download the entire blockchain. Not everyone has to run a full node, and in fact, most ordinary users shouldn't. In any case, hard disks are cheap and are only going to get cheaper. 