About amd64 It's not as confusing as it sounds (will come back to this).
To answer the original question, flash is already installed as far as I know, or you can just install it afterwards with "equo install www-plugins/adobe-flash" (as root - see the wiki pages on entropy on more details on how to use the package managers). I don't know what you mean with the question "which browser does it have"... if you have the flash package installed it should work on pretty much any browser you have on sabayon. Popular choices include (in the order I "prefer" my browsers): chromium (open source project on which Google Chrome is based on), firefox, konqueror (KDE file/web browser), epiphany (gnome browser).
As for genfool's explanation: a gnome desktop will definitely have epiphany browser (at least); a kde desktop will definitely have konqueror. But a firefox shortcut should already be on your desktop by default (firefox is the most used browser that is available on Linux).
[A little bit of x86/amd history - skip if not interested]
Now the x86/amd64, which I don't see how it's related with the original question, but I'll clarify this anyway: x86 (or IA-32) got it's name to denote the family of processors which are backwards compatible with 80386,80486, Pentium and so on, which have support for 32 bit processing. If you have experienced other distros, you might have noticed one of these prefixes in their packages: i386, i586, i686. Sabayon uses i686 for it's x86 release (according to the minimal requirements on the latest release note).
The amd64 gets a bit more complicated: this is because at some point, both Intel and AMD realized that the 32 bit architecture needed an upgrade (for instance, one of the limitations I know of the 32 bit architecture is that you cannot access more than 4GB of RAM... in fact, any 32 bit operating system I know (affects both Linux and Windows versions) doesn't even see beyond 3 GB of RAM - if you have more than 3, it will report that you have only 3). In any way, Intel launched IA64 (Intel Architecture 64) with it's Itanium processors, but those only caught a relatively small margin in the server market (you can see on the Itanium page on wikipedia that their sales are way below what they originally predicted). One of the reasons IA64 did not make an impact as the new 64 bit standard is that IA64 originally had no support for old 32 bit software - it was a full new architecture. In contrast, AMD was intended as a superset of the old 32 bit architecture - it was just an extension from 32 bits to 64. Therefore, the old 32 bit software would run just as fine on new 64 processors. Since AMD's approach became much more popular, Intel has since added support for the new standard (which we now know as amd64). You might have heard about EMT64 - this is used to denote the Intel processors which support the amd64 standard (processors in this category include some models of Pentium 4, Pentium D, as well as the "Core" series).
Some Linux distributions call it "amd64", some just call it "x86_64" to avoid confusion and to underline the fact that is supports 64 bit mode for both AMD and Intel processors. As far as I know, pretty much any desktop oriented processor that came on the market since about 2003 (even some server models too - see Intel Xeon, AMD Opteron and Phenom as an example) all have the amd64 support.
I know that you are thinking now: "If almost all the hardware that is now in use today supports amd64, then why to they bother to maintain x86?". One of the reasons should be obvious: the word "almost" in the question. Second: sometimes it happens that some software have better support for 32 bit than 64 bit (one of the most noticeable examples include early versions of Adobe Flash and various hardware drivers); of course, the opposite also happens (see the RAM example mentioned above - although this is in fact a hardware limitation, not a software one... well, most of it any way - in the sense that the 3 to 4 GB is a software design problem and above 4 it's a hardware problem... I could go into details here, but I'm pretty sure I rampaged enough for the general public... just go with it for now or ask details if some of you are interested). The choice between what release is still very debatable.
PS: I know I sometimes tend to bore everyone to death with too much information. What can I say? I like a little computer history now and then. Hopefully some of you do too.
Linux user since 2004; Sabayon user since 2006