DSC PowerSeries PG9920 PowerG 915Mhz Wireless Repeater.

Can wireless repeaters extend the range from other wireless repeaters? It doesn't seem like it does

No wireless repeaters cannot be chained in that way. They can only repeat the primary signal from the panel. 

Can a repeater help with transmitting an alert signal to a wireless indoor or outdoor siren like 1/2 mile away from the host location?

It's impossible to say for certain because of building material and obstructions between the repeater and siren, plus a 1/2 mile is quite a distance. That said, Power G wireless is the best on the market now for range and a repeater will certainly extend an already very good range of the PowerG devices. With direct line of sight and no obstructions a 1/2 mile is well within the spec of PowerG, but any obstructions will reduce range considerably.

Can the repeater be used with the power g transceiver keypad? Or it needs the actual host?

Yes, it can repeat signals from the tranceivers built into a keypad or the standalone host module.

Can i use it to repeat any others 915MHz signals?

No, this only works with DSC / Qolsys Power G wireless.

From control panel(DSC Neo hs2032) host module then repeater. Does repeater to repeater send to host?

The repeater can only repeat to the Host module. A repeater cannot repeat thru another repeater so while you can use multiple repeaters on a system, all repeaters must themselves be within range of the Host module.