Software wise there's PK_Tether
http://www.pktether.com/
Regarding the cable, a Google search found this thread on another Pentax forum
https://www.pentaxforums.com/forums/22-pentax-camera-field-accessories/3816-k10d...
I've not tried it myself but those may give you a starting point.

Not sure about Android Tablets, Android had barely come out when the K10D was first announced. I'm not saying it can't be done but suspect you'd have to write your own app and understand the cable connections to get them talking, and would be at your own risk. If you succeed you could publish it for others!
Having said that, the PK_Tether page shows what looks like an Android phone in the graphic so maybe it can (but didn't think it could though, just for the PC.)

Some recent Pentax cameras can link to Android and iOS through WiFi, but the app still seems a bit flaky.

If you wish to go high end, there's the Pentax Image Transmitter 2 software that works with the 645z and Android/iOS (if you can afford the £180 for the CD!)
http://www.ricoh-imaging.co.uk/en/product/38553/body/overview/Photo-Accessories....

Also see earlier discussion here:
https://www.pentaxuser.com/forum/topic/Laptop-as-viewfinder-5850/p-0#comm_48688

https://www.pentaxuser.com/forum/topic/Instant-view-on-laptop-5483

https://www.pentaxuser.com/forum/topic/Tethered-capture-with-Pentax-K10D-8756

Perhaps someone here still has the relevant cable and software that they can pass on, but pretty sure it would be just for PC. and of course PCs have moved on too!

John asks that spam is reported, not responded to.

He wouldn't want to be using a high ISO to photograph the paintings, there's no need as they're not fast moving subjects (even if paintings are of F1 cars!) A high ISO will lead to increased noise even if not as much as a K20 might. Just mount the painting firmly, also the camera, set it on a low ISO. Set the aperture around f8-f11 with an appropriate shutter speed. Natural daylight would be the best light source reflected if necessary to keep the lighting balanced.
If daylight not possible, a couple of flash guns set at approx 45degrees to avoid reflections.

The OP may wish to know why you feel the K20 isn't capable of this type of shot?

Could you add a link to your gallery on your profile page so we can see what, if any, problems you might be encountering. Seeing actual photos would be better than guessing the problems!

Alternatively, if you have any problem photos not on the gallery, upload them to the thread here for said reason.

Hi and welcome to PU. Just wondering how are you processing the photographs? Are you relying on the camera jpegs or are you shooting raw and processing those raw files in Phtotoshop or Lightroom or something? If the in-camera jpegs there are various settings to adjust brightness, contrast, sharpening and so on so maybe your not using the most suitable. Lightroom, for instance, can adjust out those chromatic aberrations. Also which lens are you using as some are more prone to those than others? The camera itself has a setting to remove distortions with Pentax lenses providing the lens pre-dates the camera as obviously, the camera wouldn't know about more recent lenses though updating the camera to the most recent firmware may make it aware of some newer ones up to the date of the firmware which was April 2010.
http://www.ricoh-imaging.co.jp/english/support/digital/k20d_s.html
I'm sure the K20 should be able to do what you want but without knowing your experience with cameras, software and you viewing set-up and lens(es) you have it is quite hard to make any definitive recommendation. Maybe it's just your monitor not showing your photographs to their best?
Then there is the technique in taking them. Just using the onboard camera flash would never be ideal, probably causing washed out images and unwanted highlight reflections.

So can you expand on your set up, how you take them, exactly what with , how you process them, how you view them and maybe we can start to help sort it out! I wouldn't suggest that the K20 isn't capable of achieving what you want to do and changing that alone might not make a scrap of difference if it's a technique issue. I suspect that's not the answer you were expecting. Have you read any books or web pages suggesting how best to take photos of paintings?

Some web pages on the subject to help eliminate photographic errors before looking to replace your K20:
https://www.artsyshark.com/2015/04/23/mistakes-photographing-art/
https://www.artistsandillustrators.co.uk/how-to/marketing-your-art/708/how-to-ph...
http://www.artistsnetwork.com/articles/art-demos-techniques/how-to-photograph-a-...
(Sorry if those are like teaching you to suck eggs but I've no idea of your photographic expertise so please don't be offended if that's all second nature to you!)

Were the batteries not declared to the postal staff? If they had the PO may have accepted it and put an appropriate sticker and sent it surface mail. When I enquired about it while sending something else and being asked if it contained batteries, that's what they said to do. If not, what happened would have been a likely outcome.
I expect those 'destroyed' batteries are now for sale on eBay and get posted to the buyer!

As no one else has mentioned it...

Corrected – Unstable performance may rarely occur in firmware version 1.41
Quite what was unstable they don't say!

http://www.ricoh-imaging.co.jp/english/support/digital/k1.html

Hopefully it's the unstable electronic video stabilisation now having the promised sensor stabilisation option, but you'd think they'd say that if that is what they mean.

Bin the iphones and get a £50 Android from the local supermarket?
So long as the phone can make calls, does it really matter? It's a phone!
(You'll gather I'm rather anti smartphones due to their misuse by motorists and others, especially iphones due to the slave trade involved in mining the materials needed to make them.) You could always they the new one, it's only £1100.

FWIW, if photos are uploaded with the longest side 1000 pixels, the site doesn't resize them - fact! But they may get scaled by your browser software.