Until you get a better answer this may help you. As far as I understand the semi-infinite flag manifold appeared in Feigin-Frenkel's paper [1] where they weren't really defined as an algebro-geometric object but rather they constructed what morally should be called some sheaves on them. Since both $G(K)$ and $N_-(K)\cdot H(\mathcal{O})$ are ind-groups, a priori their quotient as $k$-spaces is the sheafification of the quotient functor and it is not clear if it has a stratification by finite type schemes. I use the words *finite type* here because these spaces were introduced in the hopes that some category of perverse sheaves on them would be equivalent to some category of representations of the affine algebra for $g$ (generalizing Beilinson-Bernstein localization)(as Alexander Braverman noted below, we could deal with finite codimention strata as well). I quote from [2] (slightly different setup than yours)

*...Since the pioneering
work of Feigin and Frenkel [1], people were trying to develop the theory of perverse
sheaves (constructible with respect to a given stratification) on G((t))/B((t)) ...
The problem is that G((t))/B((t)) is very essentially infinite-dimensional, so that
the conventional theory of perverse sheaves, defined for schemes of finite type, was not
applicable. Since it was (and still is) not clear whether there exists a direct definition
of perverse sheaves on G((t))/B((t))...*

And similar quote from [3]

*The semi-infinite flag manifold, thought of as $G(K )/N (K ) · T (\mathcal{O} )$, does not carry
an algebro-geometric structure that would allow for the theory of perverse sheaves, or
D-modules, in the way it is known today.*

So it is my understanding that there has been work in trying to find inductive systems of schemes without success, in the case of the affine Grassmanian the situation is simplified because the $G(\mathcal{O})$ orbits are finite dimensional and there is a lattice model for the $GL(n)$ case. The general case and the affine flag case (not semi-infinite, but quotient by Iwahori) can be reduced to this $GL(n)$ case. In the semi-infinite situation, all orbits are infinite dimensional. The above mentioned article [3] in fact does construct a category that has all the properties that we would like perverse sheaves on $X$ to have, to that end the authors use an actual ind-scheme that serves to approximate $X$ (they use $Bun N$).

As for your second question regarding maps from a curve (the punctured disk) to the flag manifold, I'll refer you to section 3 and 4 chapter 1 (arxiv version) of [4] which by the way is the first attempt (as far as I know) of writing a category of perverse sheaves on these spaces. There's also a discussion there on the ind-scheme structures of related spaces constructed globally on a curve.

[1] B. Feigin, E. Frenkel, Affine Kac-Moody algebras and semi-infinite flag manifolds, Comm. Math. Phys. 128 (1990) 161–189.

[2] Braverman, A. Finkelberg, M.; Gaitsgory, D. Mirković, I. Iersection cohomology of Drinfeld's compactifications.
Selecta Math. (N.S.) 8 (2002), no. 3, 381–418.

[3] Arkhipov, S. Braverman, A. Bezrukavnikov, R. Gaitsgory, D. Mirković, I.
Modules over the small quantum group and semi-infinite flag manifold.
Transform. Groups 10 (2005), no. 3-4, 279–362.

[4] Finkelberg, Michael; Mirković, Ivan
Semi-infinite flags. I. Case of global curve P1. Differential topology, infinite-dimensional Lie algebras, and applications, 81–112,
Amer. Math. Soc. Transl. Ser. 2, 194, Amer. Math. Soc., Providence, RI, 1999.