In the this sentence,
Can I change it to;
Can I do this every time? If the context is clear then can I just say (sth.) 的？
Also does the second sentence sound natural (if it's grammatically correct)?
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You‘re translating 'mine' as 我的, but this jars on Chinese ears.
It seems, Chinese doesn't have 'mine' as a possessive pronoun, as in 我的家 reduced to 我的， even though 我的 will get translated as 'mine' often.
All the shortcuts Chinese makes, but it doesn't like this anaphor!!
After class I can come to your house or you can come to mine, whatever.
A friend of ours/mine
These sentences are acceptable to Chinese ears:
A very interesting question!
The sentence is unnatural because 我去你家 doesn't have 的. It's natural to say 下班以后你回你的家，我回我的.
This sentence is a bit improvement from your second one.
However, the 的 is omissible here. In speech, 我家, 你家, 他家 are common. People usually won't bother to say 你的家，我的家，他的家 because they sound wordy though they are grammatically correct.