Well yes you can indeed carry a handgun during firearms deer season. But not during archery as salukipv1 posted! You may also not carry a handgun when hunting the muzzleloader only season. You mentioned a .45 which would only be legal in a revolver and not an automatic.
Here's some info on legal handguns for deer hinting in IL:
Centerfire revolvers or centerfire single-shot
handguns of .30 caliber or larger with a minimum
barrel length of 4 inches.
And legal handgun ammo for IL:
For handguns, a bottleneck centerfire cartridge
of .30 caliber or larger with a case length not exceeding
1.4 inches, or a straight-walled centerfire
cartridge of .30 caliber or larger, both of
which must be available as a factory load with
the published ballistic tables of themanufacturer
showing a capability of at least 500 foot pounds
of energy at the muzzle.Note:There is no case
length limit for straight-walled cartridges.
Now those tracks you posted sure look interesting to me. I can't see the track imprint itself enough to say anything. But I don't believe a dog would usually walk on top of a fallen tree like that. And it looks like whtever was there kinda lept up onto the log from the spacing of the tracks showing very good balance.
I have no doubts that there are cougars in IL. And alot more than people may believe. They are rather secretive and don't roam open fields so not seeing them much isn't anything unusual. They don't put themselves on display. In the next 5-10 years I think peoples opinions will change quite a bit as evidence stacks up.
No black cats? Yep. And no cats in Il either.........