Of course every pseudo functor is a lax 2-functor, so your claiming that it cannot be a lax 2-limit (without changing its weighting)?

After your remark about the comma category (which contains the notion of slice category), it occurred to me that there is a natural application of lax 2-colimits in the bicategory of topoi, viz., the famous "Medaille en Or" exercise from SGA4.1, Exp. IV, exercice I.4.10, one major application of which is to relate the sheaves on the crystalline site to the Zariski localisations of suitable thickenings (cf. Berthelot's thesis, chp. III). Cisinski and Baez also discussed the relation of the petit/gros topos in 2009 on the n-category cafe also.

Yes it is; it's a weighted lax 2-limit.

When $n=1$, then the topology on A* = {GL}_{1}(A) is not necessarily the subspace topology (e.g. adeles/ideles). This was intentionally left ambiguous in the question. What would happen then? Your answer works when ${GL}_{n}(A)$ has the subspace topology from $M_n(A)$.