You are correct about antenna gain. The FCC spells out many of the test conditions.
I cannot answer your question about RF modems without much more study of the regulations. Most likely they are manufactured to meet some other part of the regulations and have probably been submitted to the FCC for approval. In some of these cases, it is illegal for the end user to tamper with any part of the device, including attaching an external antenna.
Presumably you mean 13 nW. I estimate that the signal strength would be about -115 dBm at the receiver (isotropic source in free space) at 300 meters. I wouldn't try it.
That band is regulated in other parts of the rules. I do not remember the appropriate section. I am certain that higher power is permitted for those specific frequencies.
See (a) above. There are exceptions listed elsewhere in part 15. For example, they allow 100 mW input power to the final stage in a transmitter on the AM broadcast band (with not more than 3 meters of antenna, lead-in, and ground combined). In the 88-108 MHz (commercial FM) band, 250 uV/m at 3 meters is allowed. The OP asked about the 144-146 MHz band which is alloted to licensed amateur use (covered by Part 97 rules). Unlicensed operation in that band is limited to 150 uV/m at 3 meters per the above table.
It is important to wade through all applicable regulations to discover what is permitted.
Cheers to you, too. John