If I'm leading a battle going up to signings, but think the other coach is about to sign the rest of his guys and then win my battle, is there any real value in jumping in right before 8pm with some effort to try and sign the guy away? 

this is d3, so it's not realistic to dump gobs of cash on him, and I can probably find a decent replacement with the cash I have left now...  but wouldn't be able to if I went hard after the player and lost. 

If you are clearly leading, the player will sign at 11am the day after signings.  If it's close it might take till the next day.
On your 2nd question I'm not sure I totally understand.  When you say he's a about to sign the rest of his guys, do u mean he will fill all his openings? If so, just let him do that and the guy will sign with you.  If not and you know you are winning, you can still wait to see if he tries to put more money into this player and then battle if he's good enough, or go after someone else.

I believe if you succeeded in knocking them off the list that it would still just be you on the list alone for a cycle. 
e.g.
4pm-both on the list
7pm-you drop some cash and he's considering only you, does not sign
10pm-signs with you, assuming no return fire

Please correct me if I'm wrong, but I think that's how it's worked for me in the past.

And the way I understand the OP, he's saying the other team has--for example--4 open spots, and they're set to sign the other 3 at the first available cycle.  Then they could come after their 4th guy that wronoj is battling for with the rest of their budget.