A.,
Several thoughts here,...first off, she is only 3 months old, their tummies are about the size of a golf ball, the food doesn't stay in there very long.. it is normal for them to want to eat in the night for some time, they are supposed to be putting on weight and growing; and it is just the countries like the US where people try to make babies conform to adult schedules rather than the other way around. Babies who experience hunger pains can certainly scream and stiffen, and as babies have little periods of growth they will need and want to eat more often. Sometimes because of distance to the babies room, by the time you hear the baby enough, and get there, the baby is in full scale hunger pain crying and once the cycles get going it is sometimes hard to get them to stop. Think about when you cry, you know, sometimes once you start it seems like you just cry about all sorts of stuff and it is hard to stop.
Babies have needs for feeling the human body, feeling the warmth and hearing heartbeat and breathing. I don't know about you, but I am not real gung ho on sleeping alone and I am in my 50's, why do we think tiny little babies want to be left to comfort themselves. Have you tried things like co- sleeping? I used to fall asleep with my baby on my chest, I didn't sleep that soundly with her in another room (tried that) so thought I might as well sleep lightly with her nearby, where I didn't have to get up all the time and turn on the lights...sometimes if they whimpered I could just place a hand on their back, or pat them a few times and they would drift off again.
If you are afraid to try the baby actually with you, you can buy these co-sleeper beds that go right beside you bed and has one side that opens to your side, it stays open so all you have to do is slide the baby over to you.. you can sleep with your hand on the baby too,so that might be helpful. Most babies are so comforted by being near their mother, you can, if you use formula arrange the makings for bottles that you have ready to quickly put together so when the baby wakes she starts to wimper and you just pour the water into the formula and feed without getting up, turning on lights etc. Everyone gets more sleep that way and it worked wonderfully for my girls. Your baby may need more body time against you, so a sling may help in the daytime. You could wear her for most of the day to help fill her need for your presence. Some babies need to be swaddled to feel more comfortable. As for adding Rice! man I can't believe the pedi suggested that... it is just so counter indicated http://www.kellymom.com/nutrition/solids/delay-solids.html
this is a link to a good article about solids and your baby.
It may be that your baby is having difficulty with something in the formula,..have you considered trying to nurse? Hard to argue with the best unadulteraated food there is. Even though you aren't now it does not mean you couldn't. La leche League has information about starting milk again, I am pretty sure.
There are also techiques that can help babies struggling to calm..here is a link to the DVD might be worth a try...
http://www.amazon.com/Happiest-Baby-Block-Crying-Longer/d...
It's called happiest baby on the block..
Hope something in here gives you ideas or hope...