yes it comes in the box

the heart rate strap is included in the pack

Yes.

Yes, from Garmin's website:

What's in the Box:
Forerunner 305
Heart rate monitor
Garmin Training Center® software
Docking cradle
Expander strap
AC charger
USB cable
Owner's manual
Quick start guide

Yes, but the plug might not fit. You can charge via USB. I have used mine in Europe and USA

Yes but you'd need a mains adaptor for the charger.

I don't live in Japan, I live in Australia and my 305 works fine here.
You can use the 305 out of the box (once you have charged it) but to get the most out of it you should enter details about yourself such as gender, age and the weight of yourself and your bike. It's quite straight forward to set up, just follow the instructions and you only have to do it once.

Yep.
not sure if the CD in the pack is for a mac, but you just download it off the garmin site.

Garmin Training Centre software can be used with Mac OS® , If it''s not included, it can be downloaded direct from Garmin

The Garmin Training software is so basic it is rubbish! Hard to get the infomation to mean anything.There is a fantastic free Mac program called TrailRunner http://www.trailrunnerx.com/ I made a donation after using it for a short while because it is now central to my training program.
You need to also download Garmin ANT Agent to really get things talking to each other nicely but it is all wonderfully drag and drop (none of the Windows Driver's nightmares!) and linked to on the TrailRunner site.
You can download tracks from within the TrailRunner Prog
which uses GPSies.com and OpenCycleMaps, to follow on your watch (best saved as a Course in Training not as a Track) or you can save the data from each ride which is overlaid on maps. The Diary feature is full of feature too. Sort of 3 programs in one. It is constantly being up-dated and tweaked with features added when enough request are made. The creator, Berbie works very hard and it shows.
You can also start your Mac as a PC and run the PC GPS programs but why open that can of worms?!

It works fine like this (assuming you are outdoors and have sat lock). There is no bike mount included though that is extra

yes it will

Yes I use it on my bike all the time and without a cadence kit

Yes. Press Mode. Choose Settings. Enter. Choose General. Enter. Choose Units. Enter.

yes you can set it to either imperial , miles or metric ,kms

Yes - no problems with metric.

I have never had a problem (even in the extremely cold winter.) I just lick my finger and run it over the two pads inside the HR monitor and put it on straight away and start riding. The only problem I have is when I ride down one particular road near me; it always reads 240BPM! Must be some telecom cable running along there?

Seems to need a bit of water on it to start with - but nothing more than running under a tap. Once it's got that I haven't had any times where it has dropped signal from the HR strap. Although I will confess to having the HR strap reasonably tight.

Depends on the person. To enhance connectivity moisten skin.

No the cadence kit is an extra item and cost a little more.

No, it can be bought as an optional extra.

no it is not designed for swimming, tey the garmin 310 xt

i wouldn't,
its only water resistant to 1m, for 30 min.
go the 310XT.

If you leave it turned on you can yes. Assuming you set the display up to show the time.

yes

Only with EU and GBR plug types. However they are for a 1A USB charger so any ipod wall charger works just as well, or a computer.

I charge it from the USB lead from the laptop, never used the plug socket