Multi-center Taub-NUT geometry and homologically nontrivial cycles

1. The existence of normalizable harmonic 2-forms in a multi-center Taub-NUT geometry is indeed not obvious at all. The specific form of these forms may be found in Sen's "Dynamics of Multiple Kaluza-Klein Monopoles in M- and String Theory", who gives them (adapted to the notation in the question) as \begin{align} \omega_i & = \mathrm{d}\xi_i \tag{1a}\\ \xi_i & = V^{-1} V_i (\mathrm{d}x^{10} + \omega\cdot\mathrm{d}x) - \omega_i \cdot \mathrm{d}x,\tag{1b}\end{align} where $V_i = \frac{m}{x-x_i}$. Sen, in turn, cites Ruback's "The motion of Kaluza-Klein monopoles" as the origin of these formulae, where it turns out that the exterior derivative in eq. (1a) is only meant to work as the derivative on the coordinates of Taub-NUT space that are not $x^{10}$, if I am reading Ruback's notation correctly. Ruback, in turn, helpfully cites "Page, D.N.: Private communication; Yuille, A.L., PhD. Thesis, University of Combridge (1980) unpublished" as the origin of these formulae, so here the trail ends.

2. $\omega^0$ in the quote from Gubser does not "owe its existence to any particular property" because it's not associated to the basis of 2-cycles that arises from the lines between the center of the Taub-NUT monopoles - you'll note that applying Poincaré duality to these $n-1$ cycles gives us only $n-1$ compactly supported (and hence normalizable) modes. Note also that these modes are not the $\omega_i$ from eq. (1a), since although normalizable, they are not compactly supported. So what Gubser means is that $\omega^0$ is an additional normalizable mode that does not show up through Poincaré duality, while the rest of the normalizable modes have "compactly supported versions" we can see through the homological argument. Conveniently, this explains a discrepancy between the Taub-NUT version of gauge enhancement and the type IIa $D_6$-brane gauge enhancement:

If you look at the type IIa approach, you'll note that the total gauge group there is $\mathrm{U}(N) = \mathrm{SU}(N)\times\mathrm{U}(1)$, but that the Taub-NUT approach looking at the cycles only delivers $\mathrm{SU}(N)$. The additional normalizable mode not arising from the cycles is precisely the explanation for the $\mathrm{U}(1)$.