I have boarded horses all my life, full care. Flymasks were NEVER a problem and cause more money to be paid. C'mon people it takes five seconds to put one on (unless the horse is weird about its face, which I can understand her not wanting to do it)
I think you're more upset about being lied to than her not giving your horse a pasture buddy. If he has one in the evening or all night, and can see horses the rest of the time I think he'll be okay. But I do not know your horse personally so I can't fully say whether he'll be okay about that.
My older gelding is a grumpy old man, he beats up other horses. So he is turned out but shares a fence line with other horses. 4 acres with good grass is REALLY hard to find...so you may have to be flexible if you want a good place that's close by.
I would talk to the owner. Talk about space for your tack. Do you have a trailer you can keep your tack in?
Horse boarding is a public service, you have to meet the needs of the boarders who are paying you. That's how it needs to be. If she's being disrespectful to you and your horses needs then you need to leave and go to a place who will repect the business you bring in.
The barn is supposed to be a fun place for you and your horse, not a stressful environment.
Neighs from Jayke and Buby!