This question is a follow up to another question of mine, which turned out to be easy (for background on Arhangel'skii's Problem see Arhangel'skii's problem revisited). Recall that a space is linearly Lindelof if every open cover which is well ordered by containment has a countable subcover, or equivalently, if every uncountable subset has a complete accumulation point.
QUESTION 1: Is there in ZFC a Hausdorff linearly Lindelöf space with points $G_\delta$ and cardinality larger than the continuum?
Arhangel'skii and Buzyakova proved that if $X$ is a Tychonoff linearly Lindelof first-countable space then $X$ has cardinality at most continuum.
Arhangel’skii, A. V.; Buzyakova, R. Z., On linearly Lindelöf and strongly discretely Lindelöf spaces, Proc. Am. Math. Soc. 127, No. 8, 2449-2458 (1999). ZBL0930.54003.
The Tychonoff property is essential in their theorem because their argument goes through taking a Hausdorff compactification of $X$. This suggests the following question:
QUESTION 2: Is it consistent that there is a first-countable linearly Lindelof Hausdorff space of cardinality larger than the continuum?
There is no hope of finding such an example in ZFC as Angelo Bella proved that every linearly Lindelof first-countable Hausdorff space has cardinality at most continuum assuming either $2^{<\mathfrak{c}}=\mathfrak{c}$ or $\mathfrak{c} < \aleph_\omega$.
Bella, Angelo, Observations on some cardinality bounds, Topology Appl. 228, 355-362 (2017). ZBL1375.54002.
Oleg Pavlov constructed, under $MA+\mathfrak{c} > \aleph_\omega$ a linearly Lindelof first-countable non-Lindelof Tychonoff space. His example is a refinement of the topology on the Cantor set and is thus of cardinality $\mathfrak{c}$ (alternatively, under MA every linearly Lindelof first-countable Hausdorff space must have cardinality at most continuum because of Bella's result).
Another consistent example of a linearly Lindelöf non-Lindelof space with points $G_\delta$ was given by Arhangel'skii and Buzyakova, but it is also of cardinality continuum (see Example 15 of the reference below).
Pavlov, Oleg, A first countable linearly Lindelöf not Lindelöf topological space, Topology Appl. 158, No. 16, 2205-2209 (2011). ZBL1226.54024.
Arhangel’skii, A. V.; Buzyakova, R. Z., Convergence in compacta and linear Lindelöfness, Commentat. Math. Univ. Carol. 39, No. 1, 159-166 (1998). ZBL0937.54022.
