These are tough requirements, and it may not be as simple as a set of specs because the requirements are inter-related.
Dimensions: there are several standard postcard dimensions, and your Postal service may have some limits on size and aspect ratio, outside of which they may charge a premium. But this one is easy enough to research and pick the paper size you want. You may be able to use a standard card or photo size, so that's ready to use. Or you could trim from the next larger standard media size. Just check the printer specs against those dimensions.
If you want edge-to-edge photo printing, that feature is more common on photo printers. Without that feature, you will need to use a larger media size and trim it (and they sell paper for standard sized photos with micro-perf tear-off edges). So paper size may be tied to printer type.
Paper thickness: A number of printer manufacturers (including Canon), have limitations on the paper thickness their consumer-grade printers can handle. Some won't handle standard cardstock thickness. They get around that by offering their own (over-priced) media that has been specified and tested to work in their printers (coupled with a warning that if you cause any damage using out-of-spec media of another brand, it isn't covered by the warranty; that's why I stopped buying Canon printers).
So the paper thickness spec to check will be either the thickness (expressed as weight), of the media you wish to use, or a spec for the type of media you want to use that is sold by the printer manufacturer and is specified as usable with the printer. This may be further complicated by the need for special media that is weather proof.
Photo printing: If you are talking about picture postcards, you need to investigate the photo print quality, which can require seeing actual output if you're picky. The specs will get you only so far (number of ink colors, droplet size, print resolution, dye-based vs. pigment ink, archival quality, etc.). Also the capability of edge-to-edge printing at the size you want.
This will be tied to the media. There are different grades of photo paper and the results can be very different. The results will also vary in matching the ink characteristics to the paper. Some photo printer inks just don't work well with some media (probably not an issue if you stick with the printer manufacturer's media). Bottom line, specs will be only a starting point. You will need to see sample output.
Weather proof: This is the fly in the ointment. Most of the printer manufacturers offer ink and media with "archival" properties and varying degrees of waterproof. Waterproof is different from weatherproof. Few of the "waterproof" ink/media combinations will stand up to sustained exposure to water.
Aside from the permanence of the ink, many papers that aren't specially designed for the purpose will become limp or curl in high humidity or if they become wet. There is media that is unaffected by most exposure to water, but printing on it with water-based ink can be a problem.
For a very limited need for postcards, you can laminate the finished cards with thin lamination film (investigate the effects--dimensions of the extra edge, the thickness, and stiffness--on the postage). This adds time and cost, and regular postage stamps may not adhere well to the film.
Other printer types: So if you want to print your own photo postcards that will stand up to weather exposure, rough handling on Postal sorting equipment, etc., inkjet printing isn't the best starting point.
Color laser printers can produce more weather-resistant output, but the toner can scratch off and photo quality typically sucks.
Years ago, a company named Alps made an economical dye sublimation printer (the printer was economical, the ribbons were insanely expensive). Dye sub produces true photo quality and is permanent. I don't know if there is currently a consumer grade dye sub printer.
Tektronix used to sell printers using a similar technology based on wax cartridges (gorgeous, permanent output, very expensive).
There used to be an inexpensive consumer photo printer that would do 4x6 photos (might be postcard size), using thermal transfer ribbons (never saw the output, but it was supposed to be good, and permanent, and the cost per print was pricey but not insane).
Another option: Many photo processing services offer printing that would be suitable for this purpose (dye sub on weatherproof media or similar). The prints tend to be pricey, but you have no investment in a printer, ink, or media.