Short answer - not really. Fat is an essential component in any cake, and milk just isn't very fatty - about 5% for whole milk. You can make cakes with milk, but they require totally different recipes: you can't simply substitute milk for oil.
Bear in mind that you're distributing the cup of oil throughout a whole cake, so that any one slice will only have a fraction of the oil. I assume you'll also be sharing the cake with others, so you'll be 'spreading the calories' somewhat.
You should be using a neutral oil like canola (rapeseed) oil anyway, as it has a relatively mild flavour. A popular alternative these days is to substitute all but a couple of tablespoons of the oil with apple sauce (really), but this can be hit and miss and doesn't work with all recipes - it usually works best with things like quick breads (scones etc).
Of course, there's also the point that cakes are supposed to be a treat, and they'd be less of a treat without the fat, in which case why bother? If you want to be healthy, make a salad. If you want a treat, make a cake and don't worry about the contents too much.
According to Cook's Info:
Jaggery is not as sweet as white sugar, so when substituting white sugar for Jaggery use about 1/3 less. When swapping Jaggery in for sugar, use anywhere from 1/4 to 1/2 more Jaggery than was called for of sugar (if you want to keep the recipe as sweet as it was.)
That meshes with my limited experience too, so if the jaggery is ground in such a way that it will measure like the sugar you are replacing, and you are replacing 1TBS of sugar, then you would want to use somewhere in the neighborhood of 1TBS plus 1tsp of jaggery. If precision is needed (which of course doesn't really apply here, but bear with me) consider that the way jaggery and sugar measures volumetrically are unlikely to really be identical. It would be better to measure by weight, or if there is liquid in the recipe, by water displacement.
Best Answer
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):
The parts that are not sucrose are the parts that are interesting. How will they affect your recipe? What adjustments should you make?
In regards to the specific foods you have mentioned:
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.