Mother cannot stop lying to me - even about little things

Later this morning, as I was about to leave the house to go to the city, my mother asked me if I wanted to a lift into town because she was going there shortly. I acquiesced, telling her I would appreciate a lift so that I wouldn't have to wait for public transport.

Once in the car, she was just her usual antagonistic, bullish self.

I cant actually talk to her and get a straight answer from her, she always obfuscates, outright lies, evades or pretends she doesn't hear me. Or, she'll answer a completely different question, one I never even asked her! Totally frustrating to have a conversation with someone who doesn't listen to a word you say.

When we got close to the city, I asked her why she was going to the city and the conversation went like this:

Me: So what are you planning to do in the city today?
Mother: Nothing. Just walk around the place. I don't have anything to do here. I just felt like going to the city.
Me: So why are you going to the city when you are not doing anything here? (I am perplexed, as she never just "goes to the city to do nothing")
Mother: No reason, just because.

Then I notice her bag with her towels in the back of the car. She takes hew own towels to her masseuse because she wants to lie on and be covered by her own towels and not the ones used there. So then, the conversation goes like this:

Me: Why cant you just be honest with me and tell me you're going to the city to get a massage?

She proceeds to ignore me, pretending that she never heard me, because she doesn't want to admit she lied about such an inconsequential thing.  Oh, the frustration dealing with someone like her is immeasurable! She is simply unbelievable! She absolutely hates it when people lie to her, she suspects everyone of lying to her and yet, she doesn't stop lying even about minor things, not just to me, but to everyone. She has absolutely no self-awareness at all.
At that point, we reached the parking lot and I told her I was going and left the car and walked to the shops to complete some errands.

No compromise is possible with my family

Last night my mother came into the kitchen, where I was preparing my food for my dinner, to tell me I must get up at 8am the next morning, in order to leave the house so that she and my step-father can vacuum the floors.

I cannot vacuum anything because I am allergic to dust and dust mites. If I am around people who are vacuuming, my body immediately reacts with allergy symptoms almost immediately (itching, blocked nose, runny nose, pain in the sinuses/lungs and painful breathing). I can be sleeping and if they start vacuuming, I get woken up with pain, even though my door is closed, I am that sensitive (unfortunately) that I get so adversely affected.

So when they vacuum the house, I have to leave and go elsewhere. It makes absolutely no sense for me to get up that early, because the only people on public transport will be office workers, all primped and perfumed ready for work and being in an enclosed space with that much perfume will just make me more sick with the allergy symptoms and is not a good idea. I have multiple chemical sensitivities, so being around anyone with any perfumed products just causes my body to over-react with major allergy symptoms, tightening of my bronchi and just feeling itchy all over. My step-father's suggestion in the past has been to tell me to catch the public transport from one end of town to the other, just sit there and do nothing. He has no clue and doesn't want to understand or sympathise. He just calls me a lazy, good-for-nothing layabout because I am sick so much.

When my mother told me about the cleaning, I told her that it would be nice if there was some compromise and not just them telling me what they wanted without actually finding a compromise. The conversation went like this (it was a heated conversation - me begging her and her being so stubborn and refusing to see my point of view):
Mother: You have to get up at 8am tomorrow morning because we are going to vacuum the house.
Me: Look, I need to wash all my linen tomorrow and hang out my duvet and pillows so they can get some sunshine to kill all the dust mites so I cant leave that early.
Mother: Well we have to clean the house tomorrow and we cant do it in the afternoon (they could, but don't want to)
Me: Can we find a compromise where everyone is happy with the outcome?
Mother: There is no compromising with you, you just want it your way.
Me: You are telling me I must do something your way, that is not only autocratic but it is no compromise either.
Mother: Well we cant clean the house in the afternoon.
Me: So why cant you let me get up later in the morning, get my stuff done and then I can leave and you can clean the house. What's wrong with that?
Mother: There is no compromising with you, we cant clean the house in the afternoon.

Then she walked off to the sun room, which is where my step-father was and proceeded to tell him all about our argument in a very angry tone of voice, complaining vehemently about me to him and he starts yelling loudly that "we are not cleaning the house in the afternoon", like they are on a schedule, when neither of them are working and besides which, that was not what I asked!

A little while later, LayZ boy walks into the kitchen, obviously he heard the argument and starting telling me I was "cuckoo" and an "idiot" and "loser". I just ignored him because, really other than lowering myself to his level, there was nothing else I could do.

It is just so difficult to deal with these people.