Hi! So a few things

• The reason the field elements are made blurry when you're running sleep[] is because they're unlocked, so are greyed-out to indicate that they're not editable while sleep[] is running. For on-screen text that you don't want the user to be able to edit it's neatest to set the fields to "locked" (Widgets->Locked when they're highlighted), that way they won't grey out when input is locked, and users won't be able to mess with them.
• In terms of adapting the eggtimer contraption, you'll likely need to edit the prototype of the contraption, if you go File->Prototypes you should be able to get to it. Probably what you'll want to do is change the "click" event into a function you can call externally, there's a bit about how to do this in the Custom Attributes section of the doco. Then you'd probably want to call this as part of the button that switches the user to a new card.
• I would probably be inclined to roll my own code rather than using a contraption for this, to be honest. Probably I'd use a hidden slider set to "animated", which means it'll run its own "on view" function every frame. Then in the "on view" basically I'd have it increment its value by 1 each frame, and then check when it gets to greater than equal than 180 (i.e. 3*60) and go to the next card when it does. And the button to move to this card I'd set to reset the slider to 0.

Hopefully this makes sense - I'm not sure how experienced you are with programming (especially in Lil) but if you need some more help I might be able to give some code snippets when I'm more awake and able to check that they actually work and such.

Following up on Millie's post -- I made a little slider example that does what she said in the last section. This slider is animated (60 view events per second), locked (not interactive for users in Interact mode) and volatile (doesn't save the value stored in it between sessions).

%%WGT0{"w":[{"name":"timerslider","type":"slider","size":[100,25],"pos":[152,105],"locked":1,"animated":1,"volatile":1,"script":"on change val do\n\nend\n\non view do\n if me.value=180 #60 frames per second x 3#\n me.value:0\n go[\"Next\"] #change this to the name of the card you want to go to#\n else\n me.value:me.value+1\n end\nend","interval":[0,180],"style":"compact"}],"d":{}}

And, explaining the script contained in it:

on view do
if me.value=180   
me.value:0   
go["Next"]   
else  
me.value:me.value+1  
end 
end 

Or, to explain it more casually:

on every view event (60 per second on this animated widget) 
IF my stored number has reached 3 seconds (180 = 60 frames x 3) 
reset me back to 0 
then go[] to another card 
ELSE (in any other circumstance) 
make my stored number go up by 1 
end (for the if statement) 
end (for the on view do)

Technically I made a "Count Up" rather than a "Count Down" but I hope it makes sense anyway! You're welcome to use or deconstruct this example for your project.

More notes:

Probably change the go["Next"] to have the name of the card you actually want it to go to?

A live countdown timer can be a little annoying while testing your other interactive elements... But it should be fine if the rest of the card is ready.

If it's annoying while testing anyway I can explain a way to make it easier to work around while editing your project. I wanted to keep it simple for this example.

When everything works how you like.... set this timerslider widget's visibility to "None" so your players can't see it. :)