In Halwa, Milk tea, cake, Indian sweets, and Custard can white sugar be replaced with Jaggery?

Of course you can use jaggery in all of these recipes or in any recipe in lieu of sugar, but like all substitutions, they may not be perfect or one to one. The outcome and flavor profile will be slightly different.

Consider what jaggery is made of (per the infamous wikipedia):

• Up to 50% sucrose
• up to 20% invert sugars
• moisture content of up to 20%
• remainder made up of other insoluble matter such as wood ash, proteins and bagasse fibers

The parts that are not sucrose are the parts that are interesting. How will they affect your recipe? What adjustments should you make?

• Invert sugars taste sweet, but are even more hydrophyllic than sucrose, and interfere with crystal formation of the sucrose. This may be an advantage or a disadvantage depending on the recipe. Candies dependant on sugar crystalization will be trickiest to manage.
• Moisture means water, you might have to reduce longer or reduce water content elsewhere in the recipe, or it could be a non-issue.
• Other content will lead to the unique flavor profile, which may be why you want to use it in the first place

In regards to the specific foods you have mentioned:

• Gajar ka halwa -- You would have to adjust the ratios, but no reason it would not work based on how this dessert is described in the wikipedia article
• Barfi -- I don't have any personal experience with barfi to know how much it depends on crystalization versus denaturing of the milk proteins for its texture--you would have to experiment. The descriptions I have read sound like it is more about the milk.
• Ready made custard powder -- I don't know what ingredients are in the "ready made custard powder"--the result would certainly be safe. Assuming the custard powder contains proteins and starch which thicken the custard when you add it to milk or water, then I see no reason why the jaggery should not work, although the result may be somewhat less thickened.
• Milk tea - no issues other than the flavor profile
• Cake -- with appropriate adjustments (mostly for the water content) it should work fine, except possibly those made by the creaming method

Note that everything I have written is based on the science, and internet descriptions of local food items I am not familiar with. You should also consult local recipe books, and find variations which use jaggery--or the lack thereof. That should give you some idea what is common, and how recipes using jaggery vary from those that don't for the same item.

The primary difference between white sugar and jaggery is that white sugar is refined and jaggery isn't and is in a more natural state. White sugar is simply crystalised sucrose whereas jaggery is unrefined whole cane sugar which includes the molasses component that refined sugar has lost.

It's this molasses component that gives jaggery its brownish colour which refined white sugar has had removed.

Molasses gives jaggery a more caramel flavour over refined white sugar and is probably a far better, purer product to use in place of refined white sugar in the examples you give. But do bear in mind the slight caramel flavour jaggery will impart due to the molasses component.

Jaggery gives more sweetness than the white sugar. Jaggery is used for making sweetened pongal. Even it can be added to coffee instead of white sugar to get taste.