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Looking for a little clarity here.

The mines say that the model in base contact must pass an init test, if they pass then the mine is resisted, if they fail they cannot strike any blows this assault phase.

A Pfist and Thammer are "difficult and cumbersome to use to attacks with a pfist are always delivered at initiative 1(ignore initiative bonuses from special rules, wargear, etc)." The thammer entry says it "uses the same rules as a pfist. In addition..."

So. Would the model get their initiative test on their natural initiative or would they use the value of 1?

The models init is used for any init test.

The weapon just strikes at init 1 but does not modify it's owners characteristic.

So the anwer is the test is its natural init.

In the same way something that had to take a str test would roll the models str and not the double str from the powerfist.

However I think if something is wounded by a thunderhammer and reduced to init 1 then it would have to take the init test at init 1 if it doesn't say 'unmodified'

As above. The weapon, when used, strikes at I1. That doesnt mean the marine isnt the I4 he still is.