Hi Karly Rose,
I had similar problems after installing 5.01. I didn't notice missing texts or calls, but the phone would sometimes "crash" with a full charge almost immediately.
Right after I installed iOS5, I began experiencing weird problems. I noticed battery life was drastically reduced: whereas I used to go all day on a charge with iOS4, after the upgrade the phone would "crash" at about 1/2 full. If I could restart, the battery would generally show that it was on fumes. Or it would show a half-charge and crash immediatly again. I never would get a "warning," low battery" message. So things were bad. But after the iOS5.01 upgrade, all **** broke loose. I didn't really suspect the battery at first, because no one on the forums was implicating batteries, and the battery indicator always showed "full" when the phone would shut down. Over time though, I observed that ALL OF MY PROBLEMS WENT AWAY WHEN THE PHONE WAS PLUGGED IN. They only happened when operating on battery power. So try that. If it works fine when plugged in, it's definately your battery.
In hindsite, I don't believe the phone was "crashing." I think it was shutting down like it does when the battery is totally drained.
After iOS5.01, the phone would "crash" randomly. Usually when I was using an app. Sometimes while I was talking. Again, it would happen with what appeard to be a "full charge," so I still didn't suspect the battery.
I tried many of the tips on this forum, for example, the tip from Mohammed from Abu Dhabi. And I restored three times. Truthfully, after doing all that, I felt like the phone returned to "pre 5.01" behaviour. That is, iOS5 behavior, which was bad, but not as horrible as 5.01 behaviour. The shut down's would generally occur when the battery meter showed about 50%. So there was an improvement. I DO think the restores and the Mohammad trick helped with whatever 5.01 was doing. But the BIG problem was the battery.
Now that I've replaced the battery, done the Mohammad trick and restored 3 times, I'm problem-free. I considered the downgrade option, and would have done it if the new battery hadn't helped. But when I went to the mac store, the "genius" was emphatic that iOS5 IS supported on the 3gs. And I have a lot of friends who are using iOS5 just fine on their 3gs's. So the problem seemed to be MY 3gs, not ALL 3gs's. And all this pointed to the battery.