Copyrights last for the life of the author and for 70 years after that. With works that are commissioned by an individual or organization, the copyright lasts for 95 years after the work was first published. If not published it can last for 120 years after its creation. Many poems are in the public domain. This means they can be used by anyone.
Is it alright to illustrate them?
Sure. I don't see why it wouldn't be.
Is it alright if the compilation is published?
This is harder to answer. The poems from before 1900 will all be in the public domain by now, so you should be able to re-publish them without any sort of copyright issues. You'll have to check each of the remaining poems and, if they'...
I think you received some good advice. Try the traditional route first. There are novella publishers like Nouvella and Narrative that accept them. And many other journals as well. Maybe try to write 2-3 novellas that relate somehow and present that to a publisher. Your chances are pretty good for success if your writing is edited and interesting.
I'm not a publisher but I have read a whole, whole lot. So this is based on my observations.
Pictures tend to be full-page, and tend to be black-and-white with only a few exceptions.
Placement is usually one of the following:
The pictures are all in one place, usually in the middle of the book
There are pictures interspersed throughout the book, on pages ...