While I appreciate the answers posted so far, the more comprehensive of the two did not include any references. Granted, it's hard to point to a reference that says something isn't required. But, I was hoping for some kind of citation. So, I went looking for one.
I believe I've finally found the applicable requirements, in the National Electrical Code (NFPA 70).
Article 702, "Optional Standby Systems" has this to say:
702.4 Capacity and Rating
(B) System Capacity. The calculations of load on the standby source shall be made in accordance with Article 220 or by another approved method.
(1) Manual Transfer Equipment. Where manual transfer equipment is used, an optional standby system shall have adequate capacity and rating for the supply of all equipment intended to be operated at one time. The user of the optional standby system shall be permitted to select the load connected to the system. [emphasis mine]
(2) Automatic Transfer Equipment. Where automatic transfer equipment is used, an optional standby system shall comply with (2)(a) or (2)(b).
(a) Full Load. The standby source shall be capable of supplying the full load that is transferred by the automatic transfer equipment.
(b) Load Management. Where a system is employed that will automatically manage the connected load, the standby source shall have a capacity sufficient to supply the maximum load that will be connected by the load management system.
I read (B)(1) to mean that the user gets to pick the load, and thus the size of the generator. However, if an automatic transfer switch is used, then (B)(2) applies, and unless the system also automatically manages the load, the generator is required to be able to supply 100% of the actual load being transferred by the switch (calculated according to Article 220, as stipulated in the main (B) paragraph).
The referenced Article 220 is a fairly complicated document. It has two different parts that address calculation of service loads (which is what I presume applies here, since the generator is acting as the electrical service), Part III and Part IV.
Part III is the more complicated of the two, and generally involves a fine-grained approach, making assumptions about lighting loads based on building square footage, but otherwise calculating the actual loads on each branch-circuit, appliances, etc. Part IV is an "optional" method, approved as an alternative to Part III. It too is complicated, but in a coarser-grained way (e.g. it allows each 20A circuit to be assumed to carry 1500 VA).
Part IV is also where we find the alternative means of determining the service load, by measuring the actual demand:
220.87 Determining Existing Loads. The calculation of a feeder or service load for existing installations shall be permitted to use actual maximum demand to determine the existing load under all of the following conditions:
(1) The maximum demand data is available for a 1-year period.
Exception: If the maximum demand data for a 1-year period is not available, the calculated load shall be permitted to be based on the maximum demand (the highest average kilowatts reached and maintained for a 15-minute interval) continuously recorded over a minimum 30-day period using a recording ammeter or power meter connected to the highest loaded phase of the feeder or service, based on the initial loading at the start of the recording. The recording shall reflected the maximum demand of the feeder or service by being taken when the building or space is occupied and shall include by measurement or calculation the larger of the heating or cooling equipment load, and other loads that may be periodic in nature due to seasonal or similar conditions.
(2) The maximum demand at 125 percent plus the new load does not exceed the ampacity of the feeder or rating of the service.
(3) The feeder has overcurrent protection in accordance with 240.4, and the service has overload protection in accordance with 230.90.
Now, the language here all seems to relate to adding load to some service, either installing new service (i.e. where the original load is 0), or adding to existing service. But, the electrician who installed the recording ammeter on my system tells me that they use the same 125% margin for the generator sizing, as well as take into account assumed heating loads if the measurement is done during the cooling season.
TL;DR: it seems that, while not precisely accurate — at no point does the code say you have to size the generator to the existing service capacity, i.e. 320 A in my case — it is true that when an automatic transfer switch is used, some load calculation is required, and the generator size must be sufficient to meet the demand of that load.
As for the other question, about the placement of the generator, all I could find on that was another NFPA reference, this time in NFPA 37, Chapter 4:
4.1.4 Engines Located Outdoors.
126.96.36.199 Engines and, if provided, their weatherproof housings that are installed outdoors shall be loacted at least 1.5 m (5 ft) from openings in walls and at least 1.5 m (5 ft) from structures having combustible walls except as provided in 188.8.131.52.1 or 184.108.40.206.2.
220.127.116.11.1 A clearance less than 1.5 m (5 ft) shall be permitted where all portions of structures that are closer than 1.5 m (5 ft) from the engine enclosure have a fire resistance rating of at least 1 hour.
18.104.22.168.2* A clearance less than 1.5 m (5 ft) shall be permitted where it has been demonstrated through methods acceptable to the authority having jurisdiction that a fire within the enclosure will not ignite combustible structures.
I presume that Generac's claim of 18" from structures is based on some demonstration of their generator enclosures not igniting combustible structures. Why the local dealer claims that this exception applies only to the generators with air-cooled engines and not those with liquid-cooled engines, I don't know. I would guess that if I wanted to push the issue, I could point to the manufacturer's advertising and technical documents.
It does seem possible that Generac is, as they claim, the only manufacturer which has gone to the trouble of demonstrating that their enclosures are capable of containing a fire. Thus, other generators would in fact be required to be 5' from the building.
Both of these sections of the regulations are, of course, not technically my local regulations. But they are part of electrical and fire code adopted throughout the US. So it seems reasonable to assume that these codes do apply in my case.
Finally, as an aside I wondered what the consequence would be of having a generator insufficiently sized for the actual load. I did not get any concrete information on that question, but from what I've been able to glean, it seems that output voltage would start to drop, until such point as there is insufficient voltage for the alternator's field coil, causing the alternator output to drop to zero, and the generator to shut off. I haven't seen anything that indicates any damage would be done to the generator. You'd just lose power again until you went and manually got the generator running again (after, of course, turning enough of the load off that the generator could handle what remained).