In one of the Tips & Notes, they explain that because mężczyzna ends with an a, it behaves like a feminine noun (its accusative is mężczyznę for example). So here it takes a -y, just like kobieta turns into kobiety
(Native Polish speakers, feel free to correct me if I'm being mistaken)
I think Grace got misunderstood. She just wanted to simplify the explanation. But she got it wrong nevertheless: Anything else around mężczyzna takes masculine forms, like adjectives and pronouns. One should rather say that some masculine nouns like mężczyzna follow the declination pattern that is otherwise used for most of the feminine nouns.
"widzieć" on its own has only the meaning "I (can) see". Not exactly "I'm looking at", even, as that is "Patrzę na".
You can use "widzieć się" for what you mentioned, rather for one-time things, so "W poniedziałek widzę się z lekarzem" (I'm seeing the doctor on Monday, literally more like "I'm seeing each other with the doctor"), also you could go with "widzimy się z lekarzem" but that works better for friends, when it's more mutual. Like, you think about 'seeing a doctor', but does the doctor think about seeing you? Meanwhile, if you're meeting your friend Susan, then you both have this 'appointment' ;)
As for dating, you can go for "widywać się" (a 'habitual' form of 'to see'), "spotykać się" (to meet) or "umawiać się" (it's like 'agreeing on something', 'agreeing that something will happen'... I don't know how to explain it well. You can say that it works as if you were 'agreeing to meet for dates' :P)