Why is "I see them in the magazine" wrong?
Because the subject is Er – He. Apart from that, "He seems them in the magazine" should be fine.
Why is "He is seeing her ..." wrong?
Because in english you do not use continous tense with verbs used for the senses: see, hear, smell, sound, taste, seem, appear,... To know more look for "stative verbs"
This sounds like it could be a line from an 80's pop song.
MY BLOOD RUNS COLD
MY MEMORY HAS JUST BEEN SOLD
why is it not "er sieht ihr in der Zeitschrift?" wouldnt the magazine be the direct object and she be the indirect object? Deutsche Sprache, schwere Sprache :(
Whom or what does he see? What is seen?
"her" is the direct object, thus you need accusative sie.
in der Zeitschrift isn't any kind of verbal object -- it's a prepositional phrase. There, the dative is requires by the preposition in indicating a location.
Hoffentlich ist es nicht Playboy...
Shouldn't also 'Newspaper' be accepted?
No, that is (die) Zeitung.
"der Zeitschrift" is correct?
That's correct in the dative case, yes.