Directory of schools – can I do this with WordPress?

I’m working on a site which works as a directory for schools. Aside from a couple of static pages and a blog (I’m familiar with customizing these) the client wants a couple of features which I’m not sure are doable with WordPress. Maybe you can help me find out. (Let me know if this is the wrong forum.)

I need two custom post types:

• School has data like name, city, country and, most importantly, geographic location (more on that below)
• Publication has data like cover photo, title, vimeo video (optional)

I have to establish a many-to-many relationship between schools and publications. One school has many publication and there are publications which have been released by more than one school.

One required feature is a google map which lists the schools as dots. Clicking on a dot shows a link to a profile of the school where adress and publications are listed.

I do code PHP but I’m not a pro. I’d like to use WordPress because of its ease of use and because login system etc. is fully implemented. (If I coded a custom CMS I’d have to do this myself.) But I’m not sure whether WordPress (and its plugins) are powerful enough. Do you have any advice?