Are All-Weather Tires helpful?
Very - I use all-season tires myself. That's primarily a budget constraint combined with the fact that Rhode Island gets serious snow rarely. All-wheel drive + all-season tires + careful driving = acceptable.
or should I buy winter tires in winter
and summer tires in summer?.
It depends on your specific situation. My previous car (97 Integra) was a similar platform to what you're using now. I was fairly happy in that car using all-weather tires on all four wheels until I got stuck in a parking lot (high-centered on a tiny lump of snow) next to my brother in law with his all-wheel drive WRX. I chose to ditch the car rather than upgrade the tires.
Admittedly not the most economically-motivated choice ever but I've been satisfied in the long run....
If you are driving in deep snow on a regular basis (i.e., Wisconsin vs. Rhode Island), I would recommend that you invest in good winter tires.
Is it okay if I buy two winter tires
(back) and two summer tires(front)?
No. Absolutely not. Please don't do that.
Remember, your front wheels in the Civic are responsible for all acceleration, most of the braking and all of the steering. If you put summer tires on the fronts and drive in the snow, it's like buttering your feet before you try to walk across a frozen lake.
Picture a clover-leaf exit ramp on your way home from work in the winter: you turn off into a fixed radius turn and hit a patch of ice from the snow melt earlier in the day. Your steering will immediately go away (summer tires can't handle ice or snow at all). Worse, your braking will be terrible in the front, so there's now chance of transferring weight to the front to get your around the turn. Absolute best case: you'll understeer into the outside guard rail, hopefully without serious injury.