This question is distinct from other questions which ask how and when to begin Shabbat, etc. when there is no sunset. I am asking about places in which sunset occurs very late during certain times of the year. For example, in certain very northerly places, sunset occurs as late as almost 2 a.m. during certain periods. In this case, even Plag ha-Mincha is not until 11:30 p.m. Are there any dispensations for celebrating earlier in these times/places, or at least eating earlier? What about in families with young children?
We did a spend a Shabbat in Alaska last summer so I had looked into this. It appears that, when the sun actually sets at night (even late), there is no real dispensation to change things but one needs to adapt the "order of the evening" to the halachic times.
Star-K writes (here) about locations like your case (e.g., Anchorage, Stockholm, Oslo) which experience sunrise and sunset 365 days a year but which, during part of the summer, never get fully dark. Times given as an example for Anchorage on June 22.
One may daven Maariv and begin Shabbos after plag hamincha (9:42pm, one and one quarter halachic hours before sunset) but he should repeat Shema just prior to chatzos halayla (1:55am, the darkest period of time). Shabbos ends shortly after that, at chatzos halayla (2:02am). One may recite the complete havdalah (besides Borai Meoray hoaysh if it is already dawn) after chatzos halayla. Alternatively, one may recite havdalah upon arising Sunday morning (only the brochos of Borei Pri Hagafen and Hamavdil).
In our case, we had a regular dinner in the evening (with Shabbat divrei Torah), then stayed late, lit candles after plag, made kiddush, ate bread, prayed birkat hamazon right away and went to bed.