Lincoln did not suspend habeas corpus unconstitutionally. Something is not unconstitutional just because you claim it was. The south did not suspend habeas corpus in a limited sense. Habeas corpus was suspended and martial law declared throughout the south, and as Mark Neeley points out in his book military tribunals operated in every part of the confederacy for most of the war. People were jailed without habeas corpus and detained without trial. And Northern courts could issue writs in their own jurisdiction for the entire war as well.
Yes he did.
Something is not unconstitutional just because you claim it was.
No. Something is unconstitutional because it violates part of the constitution. Lincoln's suspension of habeas corpus did just that. See Ex Parte Merryman for the details.
The south did not suspend habeas corpus in a limited sense.
Yes it did. The suspension applied only to federal courts and thus was limited. See the habeas corpus act for the details.