LTF circuits and $AC^0$
I doubt that AC0 can be captured by constant depth threshold circuits, but this hasn't been proved. In the case of small weights, perhaps existing methods suffice to rule out all of AC0 being captured by depth 2 threshold circuits. See Razborov's survey for several pointers, include exponential lower bounds for functions which, unfortunately, are beyond AC0

In the case of unbounded weights, we don't have any superpolynomial lower bounds. The best known lower bound for unrestricted depth two threshold circuits due to Kane and Williams, is $\Omega(n^{3/2})$.