I have a container growing lemon tree that has had lots of blossoms this year. It now is developing tiny fruit, and some branches have more than several growing. It doesn't seem like the branch would hold that much mature fruit.
You're right, some of that tiny fruit should be removed. Here's an article about fruit-thinning in general: https://www.gardeningknowhow.com/edible/fruits/fegen/reasons-for-small-hard-fruit.htm
and one on the natural habit of lemons that you might find informative: https://www.gardeningknowhow.com/edible/fruits/lemons/lemon-tree-fruit-drop.htm
I have a lemon tree in a container and the lemons start to grow and suddenly they fall off, and this goes on and on.
Look at the watering, light, and if flowers are getting pollinated. These articles have more info: https://www.gardeningknowhow.com/edible/fruits/lemons/lemon-tree-fruit-drop.htm
https://www.gardeningknowhow.com/edible/fruits/lemons/growing-lemon-trees-containers.htm
How large a container (root pot) should be used to grow a dwarf lemon tree, specifically Meyer's lemon but others dwarf trees as well.
You will want to start with a smaller pot, and gradually up-potting every year or two, work up to just about any size you can handle. This article has more information: https://www.gardeningknowhow.com/edible/fruits/fegen/dwarf-fruit-trees-a-planting-guide-for-fruit-trees-in-containers.htm
There is no fruiting in lemon plant and less flowering and flower dropping.
If the flowers are limited and dropping, that would explain why you aren't getting much fruit.
This article should help you identify the underlying cause, and correct it: https://www.gardeningknowhow.com/edible/fruits/lemons/lemon-tree-not-fruiting.htm
I planted a lemon plant brought from nursery 6 months ago. When will it bear fruits and what are the fertilizers to be given?
Lemon trees start bearing at 3 - 5 years (after you plant them.) These articles should be helpful to you: https://www.gardeningknowhow.com/edible/fruits/lemons/learn-about-fertilizer-for-a-lemon-tree.htm
https://www.gardeningknowhow.com/edible/fruits/lemons/lemon-tree-not-fruiting.htm
What can I do for them to get bigger?
Sometimes lemons drop immature fruit because the tree isn't strong enough to hold all the fruit. There are a few cultural reasons, too. This article should answer questions for you: https://www.gardeningknowhow.com/edible/fruits/lemons/lemon-tree-fruit-drop.htm
I'm guessing that the photo is a mistake, and has nothing to do with lemons.
I bought this house back in 1997 and it came along with this lemon/lime tree. In the first few years, it gave me about 10 limes per year. After that, I didn't see any more limes. It does have a lot of flowers every spring, but they fell off by the end of summer. I tried fertilizing with chicken manure and azalea food in late winter or so, but never succeeded. What can I do? Please advise.
First, you should determine whether it's a lemon or lime tree that you have. Lemons will mature into ripe yellow fruits, while limes are best harvested green - once yellow, they're too mature and unfit. These articles will help with reasons for non-fruiting in both: https://www.gardeningknowhow.com/edible/fruits/lime/is-lime-fruit-and-lime-blossoms-falling-off-tree-normal.htm
https://www.gardeningknowhow.com/edible/fruits/lime/reasons-and-fixes-for-lime-tree-not-producing-blossoms-or-fruit.htm
https://www.gardeningknowhow.com/edible/fruits/lemons/lemon-tree-not-fruiting.htm