Here are some answers

Best tourist but not lame places/things to check out in both places?

We'll give you a list of suggestions when you get here.  But, here's a teaser:

• Bombay:
• Taj hotel, Dhobi Ghat (pictured above), a movie theater, Gateway of India, Elephanta Island
• Nasik
• Um.  We're still researching this.  But, it will likely be "You should visit the place that our bus takes you to."

Where do I find bottled water and restrooms when out?

You can find bottled water pretty much everywhere.  Stalls, stores, shops, etc.  

Bathrooms, I suggest a coffee place. Costa Coffee, Cafe Coffee Day or Barista generally have clean, air conditioned bathrooms. 

When I am out, I will use a restroom at a place like that, or a restroom in HyperCity (like walmart), or in a five star hotel, or at home.

PLEASE make sure you put some TP in your purse or pocket.

Best breakfast/lunch/dinner/drinks spots to go to?

The options are really different between "town" (the southernmost part of Mumbai, pictured above) and in the "suburbs" (where you will be staying).

Rather than rewrite what's already been written, here are some great articles by Sharrell Cook:

• 6 Unforgettable Bars in Mumbai
• 6 Popular Mumbai Hangouts

Also, Bandra is an area near Juhu where you'll be staying.  There are LOTS of restaurants and bars there.

Is it hot as fuck there?

It's not that bad.  The last few days have been very dry (awesome) and probably in the high 80s / low 90s.  Nasik will border on cold at nights.