How do I sign up for an account to use the security app?

See here. 

Which alarm panels are supported by a serial connection by the mobile security app?

The security phone app is compatible with the following alarm panels:

1. IDS X Series
2. IDS 805
3. Paradox (MG & SP series only)
4. DSC PowerSeries
5. DSC PowerSeries Neo
6. Texecom Premier Panels (Premier 412, 816 & 832 only)
7. Risco LightSYS 2
8. SecureGuard Panel
9. Rhino 68, 816, 832
10. Orisec ZP-10, 20, 100

Do panel user codes allow the app user to see their assigned partitions on the alarm panel, and can they only arm/disarm the specific partitions assigned to their user code?

Currently, the app user can see all partitions but can only arm/disarm partitions assigned to the user panel code (excludes Texecom alarm panels). Assigning partition access on the app will be added soon.

Does any user code work to arm a DSC PowerSeries and DSC PowerSeries Neo alarm panel?

With the Quick Arm feature (not to be confused with the quick action on the app) disabled on the alarm panel, wrong user codes will NOT arm or disarm the panel. Only when the Quick Arm feature is enabled can an incorrect user code arm the DSC panel but it will NOT disarm it.