Q: How to make slide objects animate with single click?

When I play the presentation, I must click once for each animation for every object, instead of the slide building automatically.

1

you have not set the animations to automatically play:

select the objects on the slide

Inspector > Animate > Build In and select an effect

in the delivery dropdown menu select:   By Paragraph

click the  Build Order button > In the start dropdown menu select After Previous

2

the presentation will not transition between slides with effects (I chose timed dissolves) if I manually forward through them, the new slides just appear with no transition effect.

3

you have not set the slides with a transition:

select the appropriate slides in the Navigator panel (left hand side)

Inspector > Animate > Add an Effect and select the required effect

I can't use automatic transitions because the App's time limit is 60 seconds, and I need some slides to display longer.

4

If you mean the maximum display time for each slide;    that is 10 minutes not 60 seconds

if I set document transition to Normal, I must look at the screen and click to launch each slide's object effects one at a time.

5

If you want the same transition duration for each slide; go to Inspector > Document > Document and set the presentation type to Self Playing

If you want different durations for each slide; go to Inspector > Document > Document   set the presentation type to Normal, select the slide in the Navigator  and in the Inspector > Animate > Transitions    set the delay for the time the slide is to show on screen

Can't you manually advance a slideshow with automatic effects any more?

Yes read point 4

isn't a manual slide advance still supposed to trigger the transition effect with the set transition time?

Yes read point 4

All this information is covered in the online Keynote Help