Generally any phone capable of AT commands or IrMC should be supported. Also any Nokia phone using Nokia OS should work. For Symbian please check separate topic. You can check other user experiences in Gammu Phones Database.
Forget about using standard phones, they tend not to be reliable for long time connection to PC. Best option are GSM (GPRS, UMTS) terminals/modems. The best option seem to be Siemens modems (eg. ES75/MC35i/MC55i). Slightly cheaper, while still good are modems made by Huawer (eg. E160/E220/E1750/...). You can check other user experiences in Gammu Phones Database.
Short answer: Not really.
Long answer: For older phones (Symbian 9.0 and older), you can install gnapplet to phone and access data through it. However gnapplet has not yet been ported to newer versions, so you have no chance with recent phones. You can try using Series60-Remote, which works pretty well with S60 phones. Another option is using something what supports SyncML to retrieve contacts or calendar from your phone, for example OpenSync.
SMSD natively supports MySQL and PostgreSQL. However it has also support for libdbi, which provides access to wide range of database engines (eg. SQLite, MS SQL Server, Sybase, Firebird,...). Unfortunately libdbi currently does not work natively on Microsoft Windows, so you can use it only on Unix platforms.
Yes. You can use some of example interfaces distributed with gammu in
contrib directory. Or there is full featured separate interface
written in PHP called Kalkun.
The python-gammu project has been merged into Gammu recently, so you just need to grab Gammu and it includes python-gammu. Binaries for Windows are distributed separately.
There are lot of examples shipped with Gammu, you can find them in
python/examples subdirectory.
Gammu is known to run on wide range of systems. It can be compiled natively on Linux, Mac OS X, FreeBSD, OpenBSD and Microsoft Windows. It can be probably compiled also elsewhere, but nobody has yet tried. On some platforms however you might lack support for some specific kind of devices (eg. Bluetooth or USB).
You can quite often see messages sent from textual address or with some other nice looking sender number. However this needs to be done in the GSM network and it is not possible to influence this from the terminal device (phone). Usually it is set by SMSC and some network providers allow you to set this based on some contract. Alternatively you can use their SMS gateways, which also allow this functionality.
[GW]ammu