First of all caste is not invisible as you say. The discrimination starts from the names. Segregation based on caste is the way of life in India. I'm expecting may be its hard for a foreigner like you to notice but for the victims its very real and persistent everyday in their life!
Yes. Lots and lots. The easiest ones to identify is food preference. The upper castes ask you if you are a vegetarian. (Because they consider vegetarian = pure. Meat eaters are impure.)
Also caste determines my job, education, wealth, where I live, where I pray, who I marry and many more.
There are ways to find it from the ornaments and things, clothes you wear. It would be simply called 'tradition'. Caste segregation is the way of life here in India and there are so many different and easy ways to figure out another Indian's caste.
There is correlation between surnames (or the lack of one), language dialects, dietary habits and caste. But the signal here is very noisy and unreliable except for a few cases like having a well known surname.
To my knowledge there is no correlation to place of birth. While there may be statistical differences in the distribution of characteristics such as say skin colour among communities which I am unaware of, you will find a reasonable representation of all physiognomies? in each community.
Yes. Easier to ask them. Honestly, I think the only way to legitimize the delegitimization of the caste system is for everyone to declare their castes. No shadow games. It’s all out in the open.
Heck, when I moved to New Delhi, everyone thought I was Sikh because my last name is Singh, even though my last name is a common Hindu last name in general and not specific to Sikhs. So not even Indians are guaranteed to have the mapping of last name <-> demographic correct.
(Speaking of, do all Sikhs count as Kshatriyas? Even the farmers in Punjab and Haryana? So confusing...)