His poops sounds normal to me. Its totally normal for a breastfed baby to have a movement after each feeding utnil around 3-4 months, and then its normal for a breastfed baby to go up to 2 weeks with out a poop from 3-4 months until solids are intro'd! ITS NORMAL! Because breastmilk is tailor made for your baby, his body is using up everything in your milk, because his body is made to digest and use your breastmilk, its the perfect food for him and his little body. So going a week or so without a poop is totally normal for a breastfed baby, and it doesn't mean he is constipated at all, especially when it comes out and its mushy, it should look like peanut butter, mustard, yellowey and seedy.
So I think your son is fine. Rice is a binder, so that could be the cause of some of the mucous. Try oatmeal instead. It takes up to a week or more for food to leave the system, so his stools could continue to be funky if the food disagreed with him. ANd it could be coincidence that he got a minor diarrhea at the same time solids were introduced.
I would try oatmeal isntead of rice. and www.wholesomebabyfood.com is a great site for making homemade baby food. Much cheaper and healthier for your little guy!