R.J.
Your question contains a whole lot of variables. I hope this information will help.
The first thing I would suggest is calling the babys pediatrician regarding your childs defication issue, which may not be an issue at all. My daughter constantly deficated, till about 8 months old, while I was feeding her in the morning, it was just her time to go. But I would seriously keep in mind she is probably passing the prior days solid food while you feed her and you won't know that for sure because she eats the same thing every day.
Carrots are a fiberous food and may just be helping clean her colon-too well! I would definitely start adding other foods!!! Eating the same food every day can be boring and may be upsetting her stomache. The fact that she has a sore bottom tells me the food may be to acidic for her (jars have added citric acid). During the short time I used jar food, I used only Beech Nut-Gerber upset my daughter.
4-6 months is the ideal age to start solids and 10-11 months to start the transition from formula to milk. I make all my babyfood, which significantly helps with digestion. The rule of thumb for starting solids is to try one food for 3 days then try another, until you have given the baby all the stage 1 foods. When you start a new food, give about a TBSP the first day (1/3 a jar) then a little more each day. I gave my daughter 1 jar over the course of 3 days to make sure she didn't have any issues. The stage one foods are avocado, carrot, green bean, peach, pear, apple, sweet potato, bananas, and squash. Peaches and pears loosen the stool while veggies are a neutral. Applesauce and bananas are constipating. At 6 months you can introduce water and juice (juice should never be the constant though-it is useless calories and a bad habit) Apple juice is a laxative, so if your child gets bound up, give a 1/2 water 1/2 juice drink with lunch. All this information came directly from my daughters pediatrician.
Now, here is what I did. I started introducing all the stage 1 foods, including rice, barley, oatmeal and mixed grains one at a time. 1 meal per day of solids; I always did mornings so I could monitor any reactions. Once all the stage one foods have been introduced, I added a second meal a day of solids for about a month, then moved to stage 2 foods. My daughter started solids at 4 months and by 7 months was eating stage 2 foods and more. As an example, once you introduce green beans, sweet potatoes, rice, barley, oatmeal, banana, peach and pear...mix them up. Give her barley with peach and banana for breakfast or rice with green beans and sweet potato for lunch...just get creative. If you have the time to make her food, I would highly recommend it!!!
If you want more suggestions and ideas, let me know. I can definitely help you.