You can easily outfit yourself in such a way that makes your bike easier to use in the situations you mentioned, for much less than buying a new car.
1) Buy a rain suit--get one big enough to go OVER your regular clothing.
2) Get a basket, a rack, and panniers to carry your shit, or use a backpack.
3) If your bike doesn't already have fenders, you should probably get some.
4) Water resistant biking gloves are very helpful for when it's wet and cold.
5) Invest in a good wireless cyclocomputer--this way you can keep track of exactly how far you've gone and how long it took you.
I ride my bicycle in Oregon through the winter--through the hail, rain, and snow--and I'm a
huge weather wimp. But seriously, putting on the rain suit makes quite the difference in keeping yourself dry and comfortable.
Admittedly, there are some cities wherein biking is simply not practical, and it may be that you live in one of those. That's unfortunate. Here's a link to a business in Portland, Oregon that sells a variety of eco-friendly vehicles:
http://eco-motion.com/ Hopefully it can provide you with some relevant information, and more options.