[Xperia Z3 Compact Q&A, Help & Troubleshooting] Lasters:
As title says, I was JUST about to backup all my app data, contacts and stored messages from my Z3 Compact, when it started bootlooping on its own.

I did some researching since I have a basic background in Electronics repair (not to mention almost unlimited amounts of stubbornness and patience) and learned of a Serial debug test pads under the battery (the battery were a real pain in the rear to remove without ruining anything) which gave me following boot log (with IMEI numbers censored by me):

Format: Log Type - Time(microsec) - Message
Log type: B - since boot(excluding boot rom). D - delta
B - 253882 - SBL1, Start
B - 259158 - scatterload_region && ram_init, Start
D - 0 - scatterload_region && ram_init, Delta
B - 276147 - pm_device_init, Start
D - 27328 - pm_device_init, Delta
B - 303658 - boot_flash_init, Start
D - 217739 - boot_flash_init, Delta
B - 521733 - boot_config_data_table_init, Start
D - 0 - boot_config_data_table_init, Delta
B - 527863 - PBS setup, Start
D - 1159 - PBS setup, Delta
B - 533719 - sbl1_ddr_set_params, Start
B - 538355 - Pre_DDR_clock_init, Start
D - 244 - Pre_DDR_clock_init, Delta
D - 0 - sbl1_ddr_set_params, Delta
B - 551745 - pm_driver_init, Start
D - 237930 - pm_driver_init, Delta
B - 790041 - clock_init, Start
D - 183 - clock_init, Delta
B - 793152 - Image Load, Start
B - 907649 - Tz Execution, Start
D - 173392 - Tz Execution, Delta
B - 1090771 - Image Load, Start
B - 1147776 - Signal PBL to Jump to RPM FW
B - 1148050 - sbl1_wait_for_ddr_training, Start
D - 31750 - sbS1 BOOT
[120] USB init ept @ 0x7365000
[140] TA config read from GPT: 0x10 blocks of size 0x20000 @ 0x20000
S1 BOOT (1286-7314 S1_Boot_MSM8974AC_LA3.0_L_15.4)
[180] GPIO HW_ID[3:0]: [0000]
[180] soc_ver: 0x10001, pmic_ver: 3.1
[190] die_id: 0x1070712e, oem_product_id: 0x4, otp_lock: 0x155
PBA ID: 1285-0545 (4)
[200] CHG_STATUS_REG is 0x0 after pm8x41_chg_sts_get()
Startup flags: [ONKEY PRESSED]
Warmboot reason: [COLDBOOT]
Remote lock is UNSUPPORTED
Rooting status is: Not done
[340] OV: 0x80000 0x0 0x0
[ERROR @ S1/boot/src/s1boot_config_parser.c:595]:
MiscTA unit 2473 could not be read!
[ERROR @ S1/boot/src/s1boot_config_parser.c:845]:
None or incorrect vbus_pulse configuration!
[ERROR @ S1/boot/src/s1boot_config_parser.c:904]:
No variant configuration to store!
[360] USB ID: 1790
[360] OTG State is invalid
Service mode detected: [NONE]
[ERROR @ S1/boot/src/s1boot_lib_api.c:1448]:
TA read failed!
[450] S1 decisions complete, image to boot is 0.
[450] IMEI[0]: ##############
[450] Info: failed to retrieve secondary IMEI (optional) in facility 0x1 with code 0x2 (error ignored)
[1150] Using DTB entry 194/00010000/8/0 for device 194/00010001/8/0
[1160] CHG_STATUS_REG is 0x0 after pm8x41_chg_sts_get()
[1160] icon_flg is 0x0 battery_flg is 0x0 chg_presence_flg is 0x0
[1170] backlight_enable=1
[1190] Detected display: jdi novatek 720p cmd
[1310] No lp855x --> PMIC backlight
[1570] Battery is Good! go to HLOS
[1580] CHG_STATUS_REG is 0x0 after pm8x41_chg_sts_set()
[1580] cmdline: "androidboot.hardware=qcom user_debug=31 msm_rtb.filter=0x3b7"\
[1590] " ehci-hcd.park=3 androidboot.bootdevice=msm_sdcc.1 vmalloc=3"\
[1600] "00M dwc3.maximum_speed=high dwc3_msm.prop_chg_detect=Y andro"\
[1600] "idboot.emmc=true androidboot.bootloader=s1 oemandroidboot.s1"\
[1610] "boot=1286-7314_S1_Boot_MSM8974AC_LA3.0_L_15.4 androidboot.se"\
[1620] "rialno=YT9112XZ6P ta_info=1,16,256 startup=0x00000001 warmbo"\
[1620] "ot=0x00000000 oemandroidboot.imei=##############00 oemandroi"\
[1630] "dboot.phoneid=0000:##############00 oemandroidboot.security="\
[1640] "1 oemandroidboot.babe08b3=50000000 lcdid_adc=0x5D4F8 display"\
[1640] "_status=on androidboot.baseband=msm".
[1650] Updating device tree: start
[1680] Updating device tree: done
Leaving S1 BOOT
[1730] If you can read this line, S1 BOOT is done. Start the stone rolling!
[1740] booting linux @ 0x8000, ramdisk @ 0x2000000 (3754834), tags/device tree @ 0x1e00000
[1740] Continuous splash enabled, keeping panel alive.

Format: Log Type - Time(microsec) - Message
Log type: B - since boot(excluding boot rom). D - delta
B - 89883 - SBL1, Start
B - 95160 - scatterload_region && ram_init, Start
D - 0 - scatterload_region && ram_init, Delta
B - 112148 - pm_device_init, Start
D - 27328 - pm_device_init, Delta
B - 139629 - boot_flash_init, Start
D - 65117 - boot_flash_init, Delta
B - 205082 - boot_config_data_table_init, Start
D - 30 - boot_config_data_table_init, Delta
B - 211243 - PBS setup, Start
D - 1159 - PBS setup, Delta
B - 217068 - sbl1_ddr_set_params, Start
B - 221460 - Pre_DDR_clock_init, Start
D - 244 - Pre_DDR_clock_init, Delta
D - 0 - sbl1_ddr_set_params, Delta
B - 235033 - pm_driver_init, Start
D - 237930 - pm_driver_init, Delta
B - 473451 - clock_init, Start
D - 183 - clock_init, Delta
B - 579805 - Image Load, Start
B - 689178 - Tz Execution, Start
D - 32574 - Tz Execution, Delta
B - 731512 - Image Load, Start
B - 787754 - Signal PBL to Jump to RPM FW
B - 788150 - sbl1_wait_for_ddr_training, Start
D - S1 BOOT
[120] USB init ept @ 0x7365000
[140] TA config read from GPT: 0x10 blocks of size 0x20000 @ 0x20000
S1 BOOT (1286-7314 S1_Boot_MSM8974AC_LA3.0_L_15.4)
[180] GPIO HW_ID[3:0]: [0000]
[180] soc_ver: 0x10001, pmic_ver: 3.1
[190] die_id: 0x1070712e, oem_product_id: 0x4, otp_lock: 0x155
PBA ID: 1285-0545 (4)
[200] CHG_STATUS_REG is 0x0 after pm8x41_chg_sts_get()
Startup flags: [ONKEY PRESSED]
Warmboot reason: [CRASH]
Remote lock is UNSUPPORTED
Rooting status is: Not done
[330] OV: 0x80000 0x0 0x0
[ERROR @ S1/boot/src/s1boot_config_parser.c:595]:
MiscTA unit 2473 could not be read!
[ERROR @ S1/boot/src/s1boot_config_parser.c:845]:
None or incorrect vbus_pulse configuration!
[ERROR @ S1/boot/src/s1boot_config_parser.c:904]:
No variant configuration to store!
[360] USB ID: 1790
[360] OTG State is invalid
Service mode detected: [NONE]
[ERROR @ S1/boot/src/s1boot_lib_api.c:1448]:
TA read failed!
[370] S1 decisions complete, image to boot is 1.
[380] IMEI[0]: ##############
[380] Info: failed to retrieve secondary IMEI (optional) in facility 0x1 with code 0x2 (error ignored)
[ERROR @ S1/util/src/s1_elf_loader.c:422]:
File image is not an known ELF-file.
[ERROR IN FAC 0xB CODE 0x3 @ S1/util/src/s1_elf_loader.c:1501]:
ELF-Loader initialization failed. De-initializing.
[ERROR @ S1/util/src/s1_elf_loader.c:1577]:
Invalid parameter.
[ERROR IN FAC 0xB CODE 0x3 @ S1/boot/src/s1boot_elf_loader.c:707]:
ELF loading failed!
[420] Crash detected, but ramdumper not found.
[420] clean ramdump info
[430] rebooting.
Leaving S1 BOOT

As for the OS, I wasn't running a recent version since I was rooted (despite what the log says) and Sony being both the only manufacturer of appropriately sized smartphones and a huge trouble to anyone who likes to have a rooted and fully functional phone at the same time (often nerfing the camera when rooting via their official means).

So is there any way to recovering the rest of the data on the internal eMMC chip? I frankly don't care much about rescuing the phone itself, only the data, so if it comes to having to desolder the eMMC and dumping the data via the backwards compatibility interface many eMMC chips have that's identical to the SD-card protocol and a holding jig (with eventual 1.8v to 3.3v level converter) then so be it, the data is number #1 priority while making the phone itself useful again is only a nice bonus, so if data recovery isn't possible at all then I'm not gonna bother spending more energy on it.

Any useful help appreciated :)