There are vast double standards between the sexes.
The one I personally find most reprehensible is the dual standard that exists in many physically oriented professions regarding the performance metrics required to qualify for jobs including: police departments, fire departments, some EMT companies, the military, etc...
Oy... it pains me to say this....I agree with you.

I think if the standards for those jobs must be adapted so as to be sub standard in order for a women to qualify, then it puts the lives of fellow employees at danger If, however, a woman demonstrates the ability to meet the standards, she should be allow to be considered for the position and receive the same pay, but, I think this is rare.
I am 5' feet 3" 112 pounds and I know without doubt I could not perform the duties these jobs require...I know my limitations so, I do what is within my capability to do, change a tire, mow the grass, tune-up my car, fix a leaky sink. Mostly because I really hate to ask anyone to do anything I am capable of, and because generally I can't get it done when I want it done....proscratination on the part of men.
Something that does bug the crap outta me is the term 'Double Standard'. There are standards for women and standards for men, sometimes there's a need to for the different standards because indeed, there are some things men are more capable of doing because of their size and strength, just as there are things that women are more capable of doing because of their organization skills, the ability to multitask...they hold down jobs, run a house, take care of kids, provide taxi service to soccer, piano lessons, dance class, etc., take of their partners, and take care of themselves, and ya know, we're just naturally smarter! (kidding...get grip!).
The thing about the toilet seat and opening doors, puling out chairs, etc., It's really just a matter of courtesy, but I don't get riled if men do not perform these things. I am perfectly capable of putting the seat down, opening my own door and pulling out my own chair, and really, these things are not worth getting your BVDs or Frederick's of Hollywood in a bunch over.