For example, can I rewrite the following sentence:
я живу в Берлине с тысяча девятьсот семьдесят третьего года
as
я живу в Берлине с семьдесят третьего года
If so, is this considered informal?
Yes, you can (and should) use it, as long it is clear from the context, which century you are talking about.
As for the formality, it is only slightly informal meaning that you would not normally see the short form written in official documents, but officials may easily use it at press-conferences. At the same time, in written form, the long form would typically be replaced by digits. So in general you would encounter either
Я живу в Берлине с семьдесят третьего года. [oral]
or
Я живу в Берлине с 1973 года. [written]