Undecided questions
which is correct?

The correct one is: "La migración tomó mi corazón" (I don't know what you mean saying "la migra"). This is correct because "la migración" is singular and "tomó" is also singular. And "tomaron" is plural so it doesn't match with "la migración" written in singular.

I suppose that you want to say: "migration took my heart" or you could also say "migration touch my heart" (la migracióno tocó mi corazón), but this is just a suggestion. In Spain if you say that something "tomó mi corazón" it doesn't make sense. We don't use that expresión. It would be great if somebody from Latin America would say if that expression (tomó mi corazón) make sense there. I think it does, but I am not sure.

About the back, just a little detail: ¡Inmigración para todos!

If you want to say something like "they took my heart away", maybe you can use another verb like "arrebatar", to make emphasis or cause a bigger impact, "La migra arrebató mi corazón"

If you still want to use your sentence, "La migra tomó mi corazón" is correct, although it doesn't make much sense, it is something to be interpreted as anyone wishes, kind of soft. It is not used in Latin America, as mayte mentioned.

Rest is like you've been suggested: ¡Inmigración para todos!

You're right but it won't cause a great impact. It's better to say, "La migración/inmigración ganó mi corazón" or "Los inmigrantes ganarón mi corazón". The second sentence si very good if you wanna defend the immigrants.

On the back is perfect! Your Spanish is quite good! Remember two thinsg:
-"inmigración" have an accent on the "o".
-In Spanish you have to use exclamation mark in the begining and the end.

Greentins!