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In each of these tilings, the upper end converges to the north pole, the bottom end converges to the south pole, and the left and right ends are glued together. The "height" of the tiling is bounded, but the "width" is unbounded. In these constructions, the individual n-gons have small area, but still large diameter (it only takes a bounded number of them to reach from one pole to another); they are quite thin as the number of faces goes to infinity, consistent with the above result. So there are no counterexamples to Q1 of the OP arising from edge-to-edge tilings of convex polygons.

In each of these tilings, the upper end converges to the north pole, the bottom end converges to the south pole, and the left and right ends are glued together. The "height" of the tiling is bounded, but the "width" is unbounded. In these constructions, the individual n-gons have small area, but still large diameter (it only takes a bounded number of them to reach from one pole to another); they are quite thin as the number of faces goes to infinity, consistent with the above result. So there are no counterexamples to Q1 of the OP arising from edge-to-edge tilings of convex polygons. I suspect that some of the examples in the linked papers may also provide a positive answer to Q2 or Q3, though one would have to check each of the tilings in these papers separately to do so. (Perhaps the simplest thing would be to contact the authors of these papers for followup questions.)

Most of the tilings in the classification have a bounded number of faces (and hence diameter bounded below), but there are some families of tilings with unboundedly many faces, mostly based around the "earth map" tiling construction, which when unwrapped looks something like this:

The upper end converges to the north pole, the bottom end converges to the south pole, the left and right ends are glued together, and the bottom of the top half of the figure is glued to the top of the bottom half of the figure. In these constructions, the individual n-gons have small area, but still large diameter (it only takes a bounded number of them to reach from one pole to another); they are quite thin as the number of faces goes to infinity, consistent with the above result. So there are no counterexamples to Q1 of the OP arising from edge-to-edge tilings of convex polygons.

Most of the tilings in the classification have a bounded number of faces (and hence diameter bounded below), but there are some families of tilings with unboundedly many faces, mostly based around the "earth map" tiling construction, which when unwrapped looks something like one of these two figures (taken from Figure 3 of https://arxiv.org/abs/2307.11453):

In each of these tilings, the upper end converges to the north pole, the bottom end converges to the south pole, and the left and right ends are glued together. The "height" of the tiling is bounded, but the "width" is unbounded. In these constructions, the individual n-gons have small area, but still large diameter (it only takes a bounded number of them to reach from one pole to another); they are quite thin as the number of faces goes to infinity, consistent with the above result. So there are no counterexamples to Q1 of the OP arising from edge-to-edge tilings of convex polygons.

Wang, Erxiao; Yan, Min, Tilings of the sphere by congruent pentagons. I: Edge combinations (a^2b^2c) and (a^3 bc ), Adv. Math. 394, Article ID 107866, 36 p. (2022). ZBL1484.52016.

Wang, Erxiao; Yan, Min, Tilings of the sphere by congruent pentagons. II: Edge combination (a^3b^2), Adv. Math. 394, Article ID 107867, 68 p. (2022). ZBL1484.52017.

Yohji Akama, Erxiao Wang, Min Yan, Tilings of the Sphere by Congruent Pentagons III: Edge Combination a5

Hoi Ping Luk, Min Yan, Tilings of the Sphere by Congruent Pentagons IV: Edge Combination a4b

Wang, Erxiao; Yan, Min, Tilings of the sphere by congruent pentagons. I: Edge combinations $a^2b^2c$ and $a^3 bc$, Adv. Math. 394, Article ID 107866, 36 p. (2022). ZBL1484.52016.

Wang, Erxiao; Yan, Min, Tilings of the sphere by congruent pentagons. II: Edge combination $a^3b^2$, Adv. Math. 394, Article ID 107867, 68 p. (2022). ZBL1484.52017.

Yohji Akama, Erxiao Wang, Min Yan, Tilings of the Sphere by Congruent Pentagons III: Edge Combination $a^5$

Hoi Ping Luk, Min Yan, Tilings of the Sphere by Congruent Pentagons IV: Edge Combination $a^4b$