Partial answer...
Question: What antennas does JAXA use to communicate with its deep
space spacecraft?
From: https://www.isas.jaxa.jp/en/about/facilities/usuda.html
(and mentioned in the comments to the question above)
Usuda Deep Space Center was founded to transmit operation commands to
deep-space explorers and to receive observation data from explorers.
The location was chosen for its environment to receive faint signals
from explorers at great distances with less jamming such as city
noise.
The center has been in operation since October 1984.
The large parabolic antenna, the core component of the facility, has a
64m-diameter reflector and weighs 1,980 tons in total.
The center also has a 10m-diameter antenna and a
transmitting/receiving system to retrieve VLBI (Very Long Baseline
Interferometry) observation data.
Geographic Position: Long. 138o 21' 54EE, Lat. 36o 07' 54EN
Altitude: 1,456m
Area: 98,302m2
Question: Does it have a deep space network of its own?
Yes:
JAXA’s Usuda Deep Space Center, Nagano, ...has served as Japan’s only
ground station for deep space.
The Center conducts command transmission operations to deep space
probes and receives observation data from them as they fly closer to
satellites, planets, comets, and the moon.
X-band and S-band radio waves are used for communicating with space
probes. K-band will also be used in the future.
(https://global.jaxa.jp/about/centers/udsc/index.html)
Usada.. was used to communicate with Japan’s first two interplanetary
space probes, Sakigake and Suisei, which were launched to study
Halley’s Comet in 1985.
The antenna has since handled communications with the asteroid probe
Hayabusa and other deep-space probes.
It is now being used to communicate with the Venus probe Akatsuki and
the asteroid probe Hayabusa 2.
Similar huge antennas are used by the deep space networks of the USA, China, Russia, Europe, and India.
As also mentioned in the comments:
Because the 64 meter antenna is aging, JAXA is building a new antenna nearby.
This new antenna, called GREAT (Ground Station for Deep Space Exploration and Telecommunication), also known as Misasa Deep Space Station will be slightly smaller (54 meters in diameter) but have better surface accuracy and hence be capable of working at the higher Ka-band frequencies.
To secure operational capacity for JAXA to independently achieve
innovative results from deep space exploration by newly enabling the
receiving of Ka-band while continuing to cope with X-band
communication, which has been used for a long time for deep space
missions.
(GREAT/Misasa 54m CAD image)
(The other tracking stations: Okinawa, Masuda, and Katsuura are for satellite tracking and control)
Question: Does it use a mix of domestic dishes and those of other space agencies?
Yes:
In addition to the Center, a similar large antenna for tracking and
controlling space probes is possessed by other overseas space
organizations including NASA of the U.S.A. and ESA of Europe.
From: https://www.nasa.gov/directorates/heo/scan/services/missions/scan_network/akatsuki
The Deep Space Network provided communication and tracking services to
Akatsuki through its 34 meter antennas.
From: https://web.archive.org/web/20110614133812/http://deepspace.jpl.nasa.gov/technology/95_20/95-20.pdf
Like other outstandingly successful deep-space missions before and
after, the Voyager Mission to Jupiter, Saturn, Uranus, and Neptune and
their moons and rings depended on the unique capabilities of the DSN,
including the capability of arraying with non-DSN facilities at Parkes
(Australia), the VLA (New Mexico), and Usuda (Japan).
Additional reading:
Misasa Deep Space Station (MDSS)... will operate at X-band (8.4 GHz) and Ka- band (32 GHz). As part of
ongoing joint support of deep space missions amongst JAXA, ESA, and
NASA, ...are developing a united multi-agency effort to improve the
X/Ka celestial reference frame as well as a unified X/Ka terrestrial
frame to be shared by the three agencies.
The Misasa 54-m antenna enhances X/Ka sensitivity by virtue of having
an aperture area 2.5 times larger than the NASA and ESA antennas in
the network.
Misasa improves the network geometry by providing four new baselines
including the first direct north-south baseline (Japan-Australia) in
the X/Ka VLBI network which will provide optimal geometry for
improving declinations.
X/Ka (8.4/32 GHz) Celestial Frame:
Roadmap to the future.
https://syrte.obspm.fr/astro/journees2019/journees_pdf/SessionIII_1/JACOBS_Christopher.pdf
