It's broken!

 I've a new project. This is the door to our cabinet. The thing fell apart. I have to scrape out the glue, find some wood glue, and find some 3/4" finishing nails. Pulling out the rusted nails, I didn't think it would be that difficult.

First I looked up how to remove the glue. The piece has been glued previously. I tried all their suggestions: hair dryer to soften it, for example. It is old glue. I ended up scraping it off with the tool I use to put meds on JB's sore neck.

I found six 1" nails, but they won't go into the wood.  They are too long.

There is a piece missing here. The tongue and groove-style is simple. I don't know how to fix it.

The old 3/4" finishing nails have rusted and fallen out. I tried hammering in the new nails, but they at a tad long. I don't know how to replace the nails, nor how to secure the joints. 

Any ideas?