Just a couple of things...
I think she is waay too young to be giving up a nap, honestly I do. In fact I believe by the time her nap starts, she is probably overtired and can't settle for a long nap. Try to offer her a nap no later than 12.30pm, let her sleep for 1.5-2hrs and then bedtime 7ish. Nightwakings can and mostly are from overtiredness and I think this is what's occuring here. We ALL wake up at night, turn over, maybe go to the loo, have some water, look at the clock and then return back to sleep. Babies/toddlers also wake in between cycles but alot of the times they cannot return back to sleep because they are either overtired or are in pain from teething, or have been used to a prop to get to sleep, etc.....
I can see you have a very busy schedule but if possible try and rearrange things a little so that you are back at home for lunch about 11ish, and then maybe do a longer windown routine so that she can settle better for the nap (start NO later than 12.30pm). The longer windown should help her relax for a nap if she's been busy all morning.
I think its great that she has some daddy time & there is nothing wrong with keeping her stimulated and allow her to have lots of fun, as long as before nap and bedtime things are more or less calm.
I think her "tantrums" are largely due to overtiredness. The av awake times for a toddler is 5.5-6hrs (mine is doing 6hrs before nap now and any later, trust me she suffers at night - and so do I
). So you want her to sleep about 10.5-11hrs at night and have a 1.5-2hr nap during the day (13hrs in total). If there are days where you can't get a nap in, bring the bedtime forward to 6.30pm (again to avoid overtiredness). Most toddlers start giving up naps close to 3.
hth
Layla