The OZ flight from LHR - ICN - UUS means an overnight in ICN on both the outbound and return sectors. This arrangement was fine for my purposes and the onboard service fine. Plenty of food onboard so you get a decent meal before you arrive in Sakhalin in the late afternoon. The overnight connection might also lend itself to using two-separate tickets if you wanted to risk it (might not save you any cash in the end though).

Recent experience:
UUS airport is fairly small, but close to the city (10-15 mins on the bus). Lots of buss wait outside for the route as it would appear to be the terminus.

When you get off the aircraft you walk across the apron; if you've taken the OZ flight from ICN, you'd best get to the front of the crowd so you don't have to wait at immigration too long (only two immigration officers).

Smallest baggage belt and hall I've ever seen.
They do process the *A priority luggage first (comes off in one go - no idea when the non-priority baggage comes through).

You then have a small room for customs (you need to show your luggage labels to verify your bags are correct) and put your items through the x-ray machines. Customs officials shrug their shoulders when you asked them to stamp the customs forms.

After this a walk through a very crowded arrivals room where the local's waiting for their families to return are pretty much blocking the path.

I recall as sign for an ATM if you need it by YMMV as to whether it works.
Not found anywhere that accepts credit cards, even the local travel agencies.

I was one of the first passengers off the aircraft and through the arrivals process. Took about 15 - 20 minutes and I was already heading into town. No idea how long it would take if you were further back; no places to sit so it could be an uncomfortable wait. At least there is only one international flight arriving at a time though.

No experience with UN or SU. I noticed each had a non-soviet era aircraft parked on the apron at UUS (in fact one of them was being pushed back ready for departure). No experience traveling from the east either, but personally I'd assume OZ is your best choice. I think the Vladivostock Avia flight would be indirect and I have a feeling I some avoid-do not use advice from some official channel recently.

Note that your experience will depend heavily on when you are traveling; deep snowy winter, muddy snowy spring, dry-ish summer or muddy autumn?