1) Does CoinBase have an interest in keeping my BitCoin's safe? They do.
2) Is it competent to keep my BitCoins safe? They seem about 100% more competent than Mt. Gox, which isn't saying much, but unlike Gox I actually believe CoinBase has had its cold storage audited, especially by its rather well-known backers who almost certainly do not want to lose prestige to a massive BitCoin heist/loss...
3) If I keep my BitCoins on my home machine will they will be a) harder to access from anywhere I wish to access them and b) equally susceptible to being lost?
Yes, and yes.
I've seen too many nightmares of people losing their private keys to believe that I'm fundmentally more capable at keeping my BitCoins safe than CoinBase is.
The worst thing about this is that you are probably recommending this scheme to everyone else.
The promise of BitCoin is that it functions like cash, with all the advantages of electronic payments. If I can't bring my cash with me everywhere, then yes, it does fundamentally limit its utility to me.
BitCoin is just like any other money: the base system is based on distrust but for it to be mainstream an ecosystem of trust must be built around it. If we can't sort out the difference between competent BitCoin institutions and incompetent ones then we are in a world of trouble.
Speaking of ecosystem, I noticed that FDIC insurance suspiciously missing.
Theoretically a 'good' Bitcoin exchange would have insurance and multi-factor auth, and some complicated internal crypto scheme to minimize the scope of breaches. I can't imagine any of the current exchanges are that advanced yet.
Any half-decent online wallet or exchange keeps the vast majority (typically ~95%) of their funds in "cold storage", usually with an n-of-m encryption scheme, which will minimize the scope of breaches, with an occasional inconvenience when withdrawals exceed projections before they can pull funds from cold storage.
Even given the malleability bug (which actually isn't the real problem: http://blog.tlrobinson.net/bitcoin-transaction-malleability-...) if Mt Gox managed to lose more than say 10% of customer funds they were doing something seriously wrong.
Cold storage is an interesting concept for an exchange, but it still seems fundamentally inferior to normal banking's procedures for undoing fraudulent transactions. N-of-m encryption schemes are nice, but they only really make sense if you have each of the keys in a different hardware crypto device. Even then, in practice I suspect all of the exchanges are set up such that 95% of their cold storage is accessible via machines that are connected to the public Internet. (I'd be interested in independent audits that show otherwise.)