Yes, add the water _and_ the rice to the mash tun.
I prefer 30 minutes on a simmer, the point is to break open the starches so the enzymes can get at them. Get all the gooey glop in the tun also.
How much depends on how far you want to dry it out, and hwo good you feel about hitting your temps.
Maker's Mark (the bourbon people) use about 70% corn, 15% wheat and 15% malted barley. The last two numbers are actually 14 and 16, but I don't remember for sure which was which. The pointis that if your temperature control is professional grade you can use a metric ****ton of rice.
First time through you probably don't want to exceed 15 or 20% of total grains by weight. That is enough rice that you will need good temp control to get it converted, but also enough to change the character of your brew compared to all barley.
Don't worry about the purists. Adjunct use is a perfectly legitimate pursuit for homebrewers. No reason for rye, oats and wheat to be OK and then corn or rice to not be OK just because a commercially successful brewery happens to use them.
What I do is get the adjunct boil rolling during my acid rest, wait a little bit and then run a protein rest in the tun while the adjuncts in the other pot convert. When the time comes I dump the whole shebang into the tun and then heat the rest of the way up for sacc. rest.
FWIW Maker's Mark starts with the corn in the tun at one temp for however many hours, cool to a lower temp and add the wheat, then cool to a lower temp (probaby near 155F), and add the malted barley last for I think the tour guide said two hours. But their tuns are 11,000 gallons each.
