Flowersinthepots my aunt has a blue badge but is not obviously disabled (heart and lung condition mean she can't walk far). She's quite strident though and goes into GREAT DETAIL about her condition, the ops she's had, how she manages day to day if anyone gives her grief! I'm sorry your dad feels at all worried or embarrassed he's no need.
The sex thing - my friend with CP. Addresses this one by saying 'I can get into more interesting positions than you can! I bet my sex life's better than yours!'
I also have mh as my main condition, mobility issues are also partially due to a car accident. The mh prejudice is abominable! I get some who understand but far too many who think I'm a 'scrounger', I've worked since I was 14 full time since 16 and was in my late 30's when I stopped working (having risked my health AGAINST advice from medics by feeling guilted into working so as not to be a 'scrounger', which has probably meant I've spent MORE time out of work than I would have if I'd simply taken enough time to recover). I recently had a neighbour give me this 
face when I was getting a taxi 'again'. Because even though he knows about my health I suspect he thinks I'm 'putting it on' and I suspect he thinks taxis are a luxury those of us on benefits aren't entitled to. There's several reasons I may be getting a taxi
I'm in pain and a jolty bus doesn't help!
My legs are weak that day and I don't trust them to get me on/off a bus even with crutches.
My anxiety is bad and as I have OCD other people touching me can be an issue.
I'm attending an appointment at either therapy or Physio and they're nowhere near bus route. Re Physio this could also be an appointment for dd who has hms
But yea a taxis a luxury not allowed if on benefits! 
Dd has already experienced discrimination as she's had kids outright saying (backed up by ignorant arse parents) that she's making it up or 'milking it' - she's actually had that quoted to her several times! I'd love to send them video of her barely being able to stand due to swollen/painful knees/ankles/hips.
The school are no help! The pe teacher she had first 3 years of high school doesn't believe hms even exists! When dd was saying she needed to stop, teacher was saying 'just push a wee bit harder' totally contrary to advice from professionals including msk consultant! this only stopped after I took pics and video to show pe teacher! Thankfully dd no longer has to take pe (she swims and walks for exercise).
As for how to stop it? Standing up for ourselves is SOMETIMES an option (but toomany NOT ALWAYS as sometimes these twats run off, sometimes you sense you'd be inviting harassment, even assault, 'cripple bashing' is a thing in some places).
Personally I love that we are SLOWLY seeing more disabled people in mainstream media. Eg the last leg, the lady without a hand on cbeebies (but then look at the shit she got online), the blind actress in emmerdale, occasional actors with Down syndrome...
But it's not enough and in most cases it's tokenism/part of a plot. I'd like to see
Newsreaders, presenters, weather announcers with disabilities
More roles in TV fiction given to disabled actors but where their disability ISN'T part of the character/plot, even in action shows the police chief or whatever could be disabled.
But then all this ALSO applies to
Race (the shocking lack of non white faces in eastenders and corrie for starters!)
Sex (roles that could equally go to women repeatedly given to men - mainly police officers/drs. And when they do cast women it's part of the plot. Also lack of lgbt roles where again 'it's just who they are' - eg the very camp Sean in corrie. )
Mental health depiction (I was stupidly pleased to see monk and Whitehall depict characters with OCD as useful contributing members of society). It's always a big plot thing, not 'just another health issue' and worse when the plot has been 'going on too long' the mh diagnosis is 'forgotten' but this also happens with addiction and mc/stillbirth.
Sorry bit of an essay on prejudice generally.
We've a long way to go!