Hello and welcome.
I don't believe any of the rugs shown are particularly old, but they certainly are in very nice condition, and are also great design.
The 12x20 is a beautiful carpet. I cannot tell you exact age, however I would guess 25 years old from Iran, however could even be from India? I would say Iran over India because the white skirt, or kilim on the fringe appears to be woven tighter than that of which one would find in an Indian carpet. The fringe appears to be cotton, and pile appears to be wool. I think the KPSI is every bit of that 12x12, maybe even more. The rug is probably a Tabriz/Herati from Northwest Persia. This is also sometimes termed a Mahi design which means "fish". If you are looking for this type of carpet online, I would go for these key terms "Mahi" "Tabriz", "Herati Design". The rug does appear to be of a finer quality, and certainly is very handsome. I've always liked this design. Especially nice about this particular piece is the delicate care of design execution when it came to the samovars in the border. Very nice. You may want to consider a "bidjar" design as well, although when they have the mahi allover pattern they sometimes have medallions... For a nicer quality piece, like this, expect to spend somewhere between $55-85 per square foot in new condition.
The next rug under a piano which is red looks like a Hosseinabad. Nice design, but most likely a looser weave than the Tabriz in terms of handle and KPSI. Another search option would be "Abadeh" carpets, which are similar to this rug in regards to the highly geometric stepped spandrels in each corner of the field of the rug, however tend to have less of this repeat Herati-type field. Abadeh rugs will also often feature geometric interpretations of cypress trees running vertically in the field on opposite sides of the medallion. Search terms for this rug would be "Hamadan", "Hosseinabad", "Abadeh" with your best bet being the latter two. Be wary when using these search terms, if the price seems too good, it probably is. I'm not saying it is the case for the carpet photographed, but many of these types of carpets are now being imported by massive outlets such as Sams, BJ's, Costco, etc... Some of these newer vintage within the last dozen years or so look great, but are quite disposible in terms of quality/value. For a nice quality piece, expect to pay somewhere in the vicinity of $20-30 a square foot, even more for an older rug in new condition.
ABADEHHosseinabadThe last carpet as photographed looks like a "Jaipur" rug. These are imported from India and come in many different qualities usually made in square knots from 6/6 to 11/11 or sometimes even more. That means 6 knots horizontal, 6 knots vertical making 36 KPSI and so on. Not all Jaipurs are of this square knot, but many are. Key terms on something like this I would say, "Jaipur" would be your best bet. For a nice quality piece, expect to spend somewhere in the $20-35 square foot range in new condition around 100 KPSI.
Nice pieces, thanks for the posting!
cloudband attached the following image(s):