I have taken aware of the mealybugs infestation (thanks to BushDoctor) and I have been spraying neem oil on it. Should I do it thoroughly, like everywhere? I heard the plant can absorb neem oil and use it for themself, is that true? Anything else I should do to help it atm?
Yes, make sure you treat the tree, leaves and also the undersides of the leaves.
Also remove the fallen leaves from the soil and dispose of.
over the fall and winter, our Meyer lemon dropped all but a couple of leaves, likely due to overwatering when left outside in the rain. We pulled it inside over the winter better managing the water and early February the tree was covered in blooms. The plant remains green and flexible all over, however, the leaves have not grown back and it doesn't appear there are buds to support new growth. Is there any hope?
This article will help you.
https://www.gardeningknowhow.com/edible/fruits/citrus/leaves-falling-off-citrus-tree.htm
Newly planted, had some frosty nights. All of the have lost their leaves. Will the trees be ok, or should I re-plant?
It depends on how cold and how long they were exposed.
Only time will tell.
How many times should I spray the insecticide? When can I expect it to get better? Can egg shells help it with nutrition? It's currently battling a mealybugs infestation
You can spray with Neem Oil every 3 days for 2 weeks.
https://www.gardeningknowhow.com/plant-problems/pests/insects/mealybugs-control.htm
My mum has watered my lemon tree with banana water approx a week after its previous watering, I have noticed that some of the leaves have slightly began to cup and i’m worried. Could it be nitrogen toxicity underwatering or overwatering?
No. These need much more food than just banana water. This is likely a mix of a few issues. A lack of light, if you can't leave it outside for the winter, or if it doesn't receive 8 (6 at the least) hours of direct, unobstructed sun (windows are still a slight obstruction, and the more light the better) then it will not be enough light.
It will be happy under a 200 watt horticultural light when it is not outside.
Overwatering can cause this as well. The soil needs to dry out pretty well between waterings. It should be bone dry down to 3 inches before considering watering again.
And underfeeding. This will need a granular 3 (once at the very least) times per year, or a liquid feed every month.
This article will help with the care of lemon trees in container: https://www.gardeningknowhow.com/edible/fruits/citrus/growing-citrus-trees-indoors.htm
So recently I have been spraying my lemon tree with fungicide due to an infestation, but it seems that it's leaving white spots on the leaves. Any help?
Make sure you read the application directions on the package.
Apply fungicide in the morning, before the hot sun hits the plant.
https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm
You may want to try Neem Oil for an infestation.
https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm
Are these spider mites?
I used a solution consisting of:
48ml of water,
Half a teaspoon of vegetable oil and a quarter of a teaspoon of carex hand soap
Yes!
Treat with Neem Oil.
https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm