I have lost several lollies to rotten, why are they rotting?
Whether your bulbs are in a container or in the soil, make sure your soil is well draining.
Do not water overhead or directly into the center of the plant.
Overwatering is the most likely cause of bulb or root rot.
This link will refresh you on the care of Calla Lily plants.
what makes them weep?
This is called Guttation and it is the normal process of the plant removing excess water from the plant.
Here is an article that explains this process.
https://www.gardeningknowhow.com/houseplants/hpgen/guttation-in-plants.htm
Some of the leaves are turning brown and the plant is starting to droop even with watering
Browning flowers may be that they are at the end of their life cycle and can be removed from the plant.
Drooping plants can be due to watering issues, too much or too little.
Make sure the soil is well draining.
These links will help you.
https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/deadheading-calla-lily-plants.htm
https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/drooping-calla-lilies.htm
I received the above as a gift and noticed that some of the flowers had dark centers and some had white centers. Do these plants have male/female flowers?
Each Calla Lily has both the male and female parts of the flower. The color difference in your flowers are just that, color variations.
Here is a link that explains this flowers biology.
https://bioweb.uwlax.edu/bio203/s2013/pratt_cass/reproduction.htm
It looks like seed pods on spent flower stems. What are these and what to do with them?
Here is an article with information for you.
https://www.gardeningknowhow.com/ornamental/bulbs/calla-lily/calla-lily-seed-info.htm
Does it damage the plant to simply "pop off" the dead/dying flowers. By pulling on the flower stem it comes off clean, below the surface.
You can gently remove the spent flowers but if they resist use a snip or knife to remove them; to avoid damaging the crown of the plant.
My Calla Lilly plants came back this year and I noticed a small growth of tiny leaves next to it. I thought nothing of it at first but then I found the same thing next to another one in a different spot. I didn't want to pluck is as it was tiny and didn't look like a weed and appeared to be part of the plant however it came up next to the calla lilly. I'm just asking out of curiously. If anyone else has noticed this or if it is just my obscure observation.
It looks like you have an ornamental grass growing next to your Calla Lily plant.
This is not a part of your Calla Lily.