RELATIVITY

WARNING : NOT FOR JEE ASPIRANTS as of now

I was reading about special relativity and got some doubts
I wud be glad for someone to clear these

DOUBT 1 : TIME DILATION

it is derived t = t' / √ 1 - (v/c)2 taking a ray of light moving perpendicular to direction of motion and keepin in mind C is a constant

but when we try doing the same thing by taking a ray of light moving antiparallel to direction of motion we get

t = t' /(1 - v/c )

why ? isnt it wrong

DOUBT 2 : TWIN TROUBLE ---- IS IT LEADING TO PARALLEL WORLDS ??

Acc to theory moving frames time runs slower

NOW consider trains moving in opposite direction with speed v always in straight line

Twins A and B are on the 2 diff trains

A observes : It is getting older , faster than B.

B observes : It is getting older , faster than A .

third person on platform observes
: A and B grow at same rate !

what does it imply ? Multiple worlds ?

DOUBT 3 :

Have the above 2 paradoxes already been dealt with as they seem obvious . pls provide links

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