how to trigger strobe with alarm/switch

The siren wire should go to two diodes (non striped side).

The striped side of one diode should go to the siren.

The striped side of the other diode should go to pin 85 of the relay.

One "change" you might consider is instead of hooking pin 86 to ground hook it to (-) when armed wire like mentioned above.  With the + being supplied by the siren the relay will ONLY latch when the siren is chirping AND the system is armed.  However, this may not be necesarry depending on how long it takes the strobe to start strobing.

No they don't.  At least not for consecutive days without having a dead battery.

The general acceptable 'at rest' current draw from an automtotive electrical system is 50mA.  Most stock cars will be below 25mA at idle.

A single relay will be around 150mA.  Add that to the stock load of 25mA and you are way over the acceptable amount of current.

Find out what your batterys reserve capacity is and then calculate how long it will take to drain it with a 175mA load - it will surprise you.

In all my years of installing alarms I NEVER installed a starter/fuel pump/ignition kill that was always activated when the alarm was armed.  I rewired a few here and there that were wired that way after customers complained of dead batteries.  Every effective starter kill system I've seen connects to the (-) when armed signal for ground and the ignition for (+).  So, in reality, they are only actually interupting the starter when the ignition is on.

Here is a thread talking about reserve capacity and how long a vehicle can sit with certain loads: https://www.the12volt.com/installbay/forum_posts.asp?tid=57224&get=last