I do not mean to seem unkind on your first post, but it is said out of concern. Connecting a scope probe input directly across the mains was a really stupid thing to do! You are not the first newbie to do it though, the mains seems like a magnet to some people
! You should feel very lucky that you didn't do
yourself serious harm! Equipment is replaceable, you are not.
Please refer to this video of Dave's to understand what you did...
https://youtu.be/xaELqAo4kkQThe same bang would have occurred whatever scope you used. You were unlucky enough to be using a USB one, which extended the damage to the PC too. The USB isolated version (the VDS1022I) would have protected the scope and the PC from damage, but you would still have been at serious risk of electric shock, shorting the mains with your ground clip probably saved you from that. From your list, it sounds like an expensive lesson.
It's very hard to predict what on your scope has been damaged - the mains current will have passed through the ground clip of your scope probe, through the circuit board and via the ground signal of the USB interface to your PC motherboard and the PC Earth connection.
Given the level of damage to the PC, I would expect a similar level of damage to the VDS1022. I would suspect that the chip associated with the USB interface has also been destroyed. I don't know what you are measuring when you say "Now on board side it shows 0.3 ohm." or whether you have replaced the exploded resistors with the correct values. There is no published schematic, so it is difficult to specify individual component IDs that might be affected (I do not have the same board Rev as you). Maybe you can post a photo showing those resistor locations.
If you check that none of the pins of the USB connector (apart from the ground pin) are shorted to ground, then you might try connecting it to the PC and seeing if it is recognized. PC USB ports are quite well protected (apart from shorting mains into them!), but unplug at the first sign of an error message. The driver will need to be installed of course.
If the VDS1022 is still recognized then it should be possible to start testing other functions using a safe signal source., eg, the probe calibration output or a cheap ebay function generator. Given the cost you have already incurred, I would recommend getting the USB isolated version (which
still won't make it safe to scope the mains, but will protect the PC from accidentally introducing large low voltage currents via the ground clips).
Edit: I have been re-reading your post and am finding it a bit confusing when you say "just bought" - are you saying that you already have it working again after damaging it with the mains short and are now having triggering problems? If so, were you having them when you first got it?