You will hear both the bare infinitive ("was buy" ) and the full inifinitive ("was to buy"). Both are acceptable. "Costly" and "expensive" are synonyms, so either word is acceptable. "Expensive" is more commonly used, but "costly" certainly isn't wrong. Here's a discussion about it:https://english.stackexchange.com/questions/197326/how-to-use-the-first-thing-i-did-was

No this sentence is not right. You have to say: The first thing he did was buy an expensive car.

While it's not incorrect, it would be more natural to say 'The first thing he did was buy a new, expensive car.'

Yes it is. Or you can say 'an expensive car'.