I transplanted the tree 1 month ago and it already has flowers and small fruit. I've heard it's a good idea to remove all the fruit in the first season to avoid stunting the growth of the tree. Should this be done in spring, summer, or fall? Thank you.
This should be done at the earliest time possible. Now will be fine, as there is no real time constraint.
these articles will help:
https://www.gardeningknowhow.com/garden-how-to/gardening-by-zone/zone-7/zone-7-citrus-trees.htm
https://www.gardeningknowhow.com/garden-how-to/gardening-by-zone/zone-8/zone-8-orange-trees.htm
https://www.gardeningknowhow.com/edible/fruits/tangerine/mandarin-orange-tree-care.htm
Dwarf Satsuma plum bought & planted 2018. Leaf drop was very late in 2019. Looks healthy but had no flowers. Mariposa plum flowered a lot.
These articles should help: Look at overfertilization and improper pruning. (Is tree getting extra nitrogen from lawn fertilization?) A soil test might provide the answer to what's missing.
https://extension.psu.edu/home-orchards-why-is-there-no-fruit-on-my-tree
https://www.gardeningknowhow.com/edible/fruits/plum/satsuma-plum-care.htm
https://www.gardeningknowhow.com/edible/fruits/plum/plum-trees-not-fruiting.htm
I have a 10+ year old Satsuma. fruit used to be sweet with thin peeling. Now peeling very thick and knobby fruit not as sweet--have I lost my graft??
Most likely. Look closely where the graft meets the desired cultivar. Does it look like branches from below the graft are overtaking the branches above the graft? You can try pruning them out if there aren't too many. Try not to remove more than 1/3 of the branch structure at one time.
This can easily happen when you don't think about checking for it. Each spring, check below the graft for any sprouting branches and remove them.
You can also check for a nutrient imbalance. A soil test will help. Your local extension office can do that for you.
https://www.gardeningknowhow.com/edible/fruits/citrus/citrus-fruit-thick-rind.htm
My tree is about 8 to 10 years old and finally, this year, it was loaded with blossoms. However, I see no signs of fruit now that the blossoms have dropped. I live in west central FL.
Depending on whether you are talking about the Orange or the Plum, there could be a few reasons for the lack of fruit production.
If this is the Satsuma Citrus that you are referring to, then it is likely a soil fertility or pH issue. Here are some articles that will help:
https://www.gardeningknowhow.com/edible/fruits/oranges/no-fruit-on-orange-trees.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/soil-ph-plants.htm
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/testing-soil.htm
If this is the plum, then it is likely that your area is too warm (or rather doesn't receive enough cold) to produce. It could, still, be a nutrient or fertilizer issue, but plums will enjoy a cold period during the winter. Here is an article for more information on why plums fail to fruit:
https://www.gardeningknowhow.com/edible/fruits/plum/plum-trees-not-fruiting.htm
It is also oozing out some liquid. Some of the bark has come off of the tree trunk. On the tree itself it looks healthy with lots of fruit on it. They usually are ripe around December/January. It has for the most part always had wonderful fruit. Can this tree be saved or do we need to get rid of it?
I will assume that this is a Satsuma citrus, not plum, due to the area in which you reside. If this is, indeed, the case then your tree is quite aged! This is, relatively, old for any citrus species.
With this being said, your tree is prone to many issues that would not be a problem for a younger, healthier tree. In this case, this is Bacterial Slime Flux. Not a deadly disease, but it does mean that your tree needs attention. Usually, a feeding will take care of the issue. Here is an article that will help:
https://www.gardeningknowhow.com/ornamental/trees/tgen/wetwood-bleeding-trees.htm
I would like to preserve the fruit from rotting on tree
You can keep them at about 32 degrees F for storing two weeks. That is a little colder than the refrigerator (40 degrees F). The freezer would be too cold at 0 degrees F.
https://www.gardeningknowhow.com/edible/fruits/plum/picking-plum-fruit.htm
The satsuma has fruit too
likely, the graft was near fruiting age when it was installed. This will be the reason for grafting, typically, so that you can get fruit much faster.
https://www.gardeningknowhow.com/edible/fruits/tangerine/mandarin-orange-tree-care.htm