people+who

both are acceptable, but 1 is preferred and more widely used.

Teo1. People who have their dreams come true are happy.

2. People that have their dreams come true are happy.

Sentence #1 is correct. Is sentence #2 also acceptable?

'that' can be used for both living and non-living things.

To have something done is not a good structure for this sentence. Better:
People whose dreams come true are happy.

But:
I'll have my hair cut next week. (= Someone else will cut it for me.)
He had his wallet stolen. (= He was the victim of a pickpocket, someone stole his wallet.)

English is sometimes a little inexact due to a lack of inflections and oth

1. 'I cut my hair' is incorrect if you mean you went to the barber's.

2. 'I had my hair cut' is correct.

3. 'I broke my arm' is correct.

English usage sometimes appears illogical.

OR

"People who / that realize their dreams are happy", if that's what's meant.

From what I could find at the New York Times, both constructions are used mostly with the possession meaning of have (I own/possess):

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http://select.nytimes.com/gst/abstract.html?res=F60A12FC3C550C768EDDAC0894DB404482