But I want to focus on countries that use pure proportional representation (open party-list PR system) in conjunction with levelling seats - such as Denmark, Sweden and Norway. Is it still possible for these countries to produce overhang seats? More importantly, how do they deal with it if it happens?
Yes, it is possible if there are not enough leveling seats compared to the fixed seats.
If a party receives more fixed seats (the seats directly awarded by each constituency) than it would be entitled to according to the national proportional distribution, those extra seats would be overhang seats. This has happened repeatedly in recent years in the Swedish Riksdag elections. The outcome is described well in the Swedish Wikipedia article about leveling seats (utjämningsmandat):
T.o.m. valet 2014, om ett parti fick färre mandat i uträkningen än de redan hade av de fasta mandaten, behöll de sina fasta mandat, och en ny totalfördelning gjordes med det partiets fasta mandat borträknade. Fr.o.m. valet 2018 återförs istället de överskjutande mandat som tilldelats med lägst jämförelsetal.
Vid valet 2010 räckte inte utjämningsmandaten och Socialdemokraterna fick 112 mandat istället för de 109 de hade fått om hela landet räknats som en valkrets. Likaså fick Moderata samlingspartiet 107 istället för 106. Missgynnade var fp, kd, v och mp, vilka alla skulle fått ett mandat mer var, om hela landet hade räknats som en valkrets.1
Vid riksdagsvalet 2014 skedde detta igen. Denna gången gynnade det Socialdemokraterna (+1) och Sverigedemokraterna (+2), medan Moderaterna, Folkpartiet och Kristdemokraterna alla fick 1 mandat mindre än om hela landet räknats som en valkrets.
Up until 2014, if a party received less seats proportionally than they had already been awarded in terms of fixed seats, they were allowed to keep their fixed seats and a new proportional distribution was made with the overhang seats subtracted. Starting from 2018 the overhang seats are instead subtracted from the party's fixed seats to ensure proportionality.
In the 2010 election there were not enough leveling seats and the Social Democratic Party received 112 seats instead of the 109 they would have received proportionally. Similarly the Moderate Party received 107 seats instead of 106. The disadvantaged parties were the Liberals, the Christian Democrats, the Left Party and the Green Party, all of which would have received one extra seat if they were distributed proportionally.
In the 2014 election this happened again. This time the beneficiaries were the Social Democratic Party (+1) and the Swedish Democrats (+2), while the Moderate Party, the Liberals and the Christian Democrats all received 1 seat less than they would have received proportionally.
The Swedish system was not designed for as many parties as the Riksdag currently has, so there are too few leveling seats. This became apparent in 2010 and 2014 after the Swedish Democrats had passed the election threshold. Hence the rules were changed for the 2018 election to ensure proportionality. Using the classification system in the Wikipedia article on overhang seats, Sweden changed from method 2 to method 3. An alternative solution would have been to simply reduce the number of fixed seats and increase the number of leveling seats.
No, this is not possible.
It is not possible to create overhang seats in the system used currently by Denmark, Sweden and Norway (among others).
The way seats are allocated is in a strictly descending order of progressive quotients which is based on the number of votes first in the individual ten constituencies, then in the aggregated three provinces.
Effectively, the Danish (and similar) system(s) use only one vote to award one type of seat by giving the "next seat" to "the next party in line", first with the constituency seats and then with the national leveling seats. Parties have no way to "skip the line" and earn extra seats first.
In contrast, mixed-member proportional systems (MMP) (as used in e.g. New Zealand, Germany) have two different votes to award what are for election purposes two different kinds of seats, and they are also decided in different ways (constituency seats are given absolutely, list seats are given proportionally). Because no check is made on the total number of seats arising from the two stages separately, this total can exceed the nominative size of the e.g. Parliament, Bundestag resulting in either acceptance of excess representation or a corrective process to maintain proportionality.