Am considering a very full day trip to Springfield, IL (would stay overnight to get a really early start here, as there's a lot of small stuff and one larger attraction). Feedback wanted, especially as to whether the Oliver Parks Museum is closed or not. On a map, these all seem to be pretty close by each other, which would help keep the approximate timing possible.

8:30 AM--get tickets for Lincoln house tour at visitor center, for an after 5 PM time
8:45 AM--State Capitol
9:30 AM--Executive Mansion tour
10:00 AM--Grand Army of the Republic Museum
11:00 AM--Oliver Parks Telephone Museum (this would be the one to jettison if need be)
11:30 AM--Old State Capitol
12:30 PM--Dana Thomas House
2 PM--Lincoln Presidential Library
after 5PM--Lincoln House tour

I'm about two hours from S'fld, but hardly ever go there. I've done the Lincoln House many years ago, as well as the old State Capitol and Dana Thomas House. I do not have any experience with your other places. Sounds like a very full day, IMO. I hope you don't run yourself too ragged.

One of these days I must get to the Lincoln Library! My friends who have been there are all very impressed! Allow for plenty of time there!

Also in Springfield: Lincoln law offices & grave/monument.

I did notice that the Dana-Thomas House is closed on Mondays and Tuesdays and closes at 4:00 p.m. on the days it is open.

I visited Lincoln's Tomb when I was in fifth grade and is still a place I have to go when I'm in Springfield. It's an almost eerie feeling knowing that he is actually buried right there. Also, the old tomb is around back of the current tomb.

45 minutes doesn't seem long enough for visit to Capital. I would give at least an hour for that.

This day seems really really busy to me. What about lunch at Cozy Dog?

If you are going just to see all the sights and not tour them, then you will have enough time. But if you want to take guided tours of these site you will need 2 days.
Cozy Dog is fun for lunch and you should try a Horseshoe sandwich, which is unique to Springfield at various restaurants and pubs in town.