"There is pain in my thumb."
Translation:मेरे अँगूठे में दर्द है ।
This is the usual way to frame the sentence in Hindi. It is similar to how a prepositional phrase can begin a sentence in English even though it is the object. Eg: In 2013, Obama was President.
'दर्द मेरे अंगुटे में है' is also a grammatically correct sentence but is not used much except as a response to the question 'Where does it hurt?'.
It's not really a set phrase.
Notice that the English sentence uses the expletive construction 'There is..' instead of the usual subject-verb-object sentence 'Pain is there in my thumb' to facilitate easier understanding and to place some more emphasis on the subject. Hindi does the same by switching the words around.
For example, मछलियाँ पानी में हैं = 'The fish are in the water' (Answers the question 'Where are the fish?')
while पानी में मछलियाँ हैं = 'There are fish in the water' (Answers the question 'What is in the water?')
Yes. Hindi is pretty flexible with word order. In fact, colloquial spoken Hindi is almost word-order free with different placements of words used to convey subtle changes in meaning. Formal Hindi is less so but there is still some leeway in switching words here and there.