CoinBase bills itself as a digital wallet before it bills itself as an an exchange. They are at least domestically based which gives me some legal recourse in the event that they do lose my coins.

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.