You are correct in your inferring.
Thanks very much for this, schokie. Was I correct in inferring that you're brewing a more concentrated wort that will be diluted with the sparge water? I'm still uncertain how to plan a recipe that would include (say) 5 gallons of wort from the mash and X# gallons from the sparge. Also, would the sparge water need to be boiled out, or can one add it straight to the carboy?
I'm not sure what you mean by sparge water. Sparge water is the extra water you use to rinse the grains prior to boiling. Sparge water is boiled. The water used to dilute the wort post-boil to ensure a specific amount in the fermenter is generally referred to as top-off water. It does not have to be boiled. Some OCD people will actually boil the top-off water, or filter it, reverse osmosis, etc. I just use plain old tap water. So far, so good!
The planning is actually quite straight forward. You just can't be afraid of fractions.
For example, dry extract normally is about 43 points per pound per gallon. So 1lb extract in five gallons is 45/1/5=9. That would be a gravity of 1.009. Obviously you want more than that, so start with your desired OG and work backwards. A 1.050 OG would require 50=43/X/5. Solve for X and you need 5.8lbs extract. Now is when you're happy you paid attention to algebra in high school. If you're using all grain, the math is the same, but assume 36 pppg instead of 43. It varies a bit depending on the type of grain, but using 36 as an average will get you close enough. Don't worry about efficiency for now. Change 5 gal to whatever size batch you want to do.
BTW, only barbarians refer to it as ppg. It is correctly called pppg. Moving on...
Now, decide how big your boil is going to be. While I can boil right up to the rim of my 10 gal kettle, that's a lot of water to lift. I normally plan on a 7.5-8.5 gal boil. If you're using extract, this part is easy. The amount of extract/grain is independent of your boil kettle volume. So in my 10 gal kettle I fill it up to about 7 gals with water, heat it up, add the calculated extract, boil, cool, and then evenly split it into 2x 5-gal fermenters. I then add top-off water to bring both fermenters up to the 5-gal mark. To be pure, they're actually 6.5gal fermenters, but I'm brewing 5 gal in each. Don't forget the krausen. The amount of top-off water isn't important. Neither is the boil volume. The only amounts that matter are the final volume into the fermenter and the amount of extract to get to my desired OG.
For all-grain, it's slightly more complicated, but the basic process is the same. I go through the math I mentioned above to calculate the amount of grain I need. I then multiply the grain lbs by 1.25 to calculate the quarts of water for the mash. 1.25 qt/lb is just an average mash density. Don't be afraid to go a bit higher, but I wouldn't go much lower. So 10lbs of grain would need 12.5 qts, or about 3+a bit gallons in the mash. So I take about 3 gallons of water, heat it to my desired mash temp, then dough in the grain. Once the mash is complete, I drain the water out into my brew kettle. I then add 4 gallons to the mash tun for a sparge (7gal boil-3gal mash). Drain that into the boil kettle and the rest is as described above. I actually also factor in mash tun deadspace and about 1/4 qt/lb of water for grain absorption.
If you're keeping track, you'll notice my process requires two kettles. I use my 10 gal kettle for the boil and I use a second 5 gal kettle to heat mash/sparge water. If you only have 1 kettle you can use the no-sparge method, but that will reduce your efficiency some.
(related question: does boiling alter the chemistry of the wort, or is it just a means of eliminating excess water that was required for the mash?)
Yes. It does both. The boil drives off volatile precursors for off flavors like DMS. 60 minutes isn't always required, but it will depend on the type of grain used. The tradition is a 60 minute boil, which is what I normally do, but there are a number of good recipes out there that use a shorter boil. With extract, even just a 15 min boil can be fine. It also concentrates the wort and reduces volume.