I think a deathly allergy would qualify as a 'legitimate safety requirement'. There are also situations in which the ADA considers an allergy itself to be a disability, though the given examples are usually around food allergies.

I don't think a deadly allergy to dogs exists as a condition on its own, you would probably also have a whole lot of terrible respiratory issues when exposed to a lot of other allergens. Someone with a deadly allergy to dogs will probably already require accommodations on the job that shield them from close interaction with people on its entirety.

Such a person would not be able to drive a taxi/uber regardless of accommodations. I used to have a severe cat allergy that resulted in an asthma attack when exposed to, for instance, a colleague covered in cat hair. Their cat was in another city at the time.

I don't see that language anywhere except for the case of miniature horses?