What's the reason for the word order of "would sooner have"?

I'm a native speaker from the UK, but after living for more than 10 years in a foreign country, I'm beginning to notice how my English is getting much worse.

The other day, I came across a phrase that I hadn't used in a long time - would sooner have. And I began to think about its past tense. I asked a neighbour who is an English teacher (it is however her second language) and she told me that have should go after sooner e.g I would sooner have done something. I then thought, why wouldn't it be also correct to say I would have sooner done something (well she told me this one was wrong). So what's the reason behind this word order?

The explanation is that "would sooner" is an idiom. This is found in a note to § 3.45 of "A comprehensive grammar of the English language". According to the analysis in this grammar the construction "would sooner" can be reckoned with as a modal idiom alongside other idioms of the same type. The additional idioms are joined to the list shown below.

3.45 CoGEL Modal idioms: had better, etc

This category contains the following four multi-word verbs, as well as some less common verbal constructions:
             had better     would rather    HAVE got to     BE to
They all begin with an auxiliary verb, and are followed by an infinitive (sometimes preceded by to):
We had better leave soon. Yes, we had. [1]
We'd better leave soon. Yes, we'd better. [1]
I'd rather not say anything. [2]
They've got to leave immediately. [3]
The conference is to take place in Athens. [4]

_{Note
[a] In addition to the four modal idioms illustrated above, the following sentences illustrate less common idioms which might be placed in the same category:
      I would sooner leave the decision to you.
      I would (just) as soon eat at home.
      We may/might (just) as well pay at once.
      You had best forget this incident.}