Chris Kitson

Try and avoid an inverter if you can (i.e. 220V)
Stick to 12V DC... This will give you far more efficiency than an inverter.

If you are living in this home full time (e.g. not just on weekends or such) then I would suggest you rather go with lithium batteries (better for all time work) instead of lead acid (better only as backup) as it should have a longer lifespan and lithium comes with a BMS to protect the battery. 
I agree with Richard. If you can get away without using an inverter there will be much less losses. The Notebook might be the difficult item in your list here. All the other items should already be low voltage DC.
For those devices that does need an inverter, you can manage when you use what devices and don't plug them in all at the same time so something small should work. (like a Victron Phoenix 300). Make sure the total watt value of all the devices you will use at the same time is less than the watt rating for your inverter. E.g. if you are going to use the notebook and 2 LED.
For the battery storage calculate the watt value for each device and how long they will be used. E.g 10 hours for the laptop @ 300w and 3 LEDs for 4 hours @ 5W to give you a total watt hours required (3060Wh) and divide that by the battery voltage for the amp hours you need (12v will give 255Ah). Lithiums you can use almost the full spec, while lead acid don't like that so you will need to rather get more lead acids so they don't drain fully.
To charge the battery you need a solar charge controller to change the battery from the solar panel. PWM or MPPT should be fine in this case with 1 panel, but you need to be able to charge your battery and 360W panel with say 6hours sun is only 2100watt or 180Ah @ 12v. 
So I would rather look at something like 600w solar that will give you 300Ah @ 12v (something like as Victron SmartSolar 100/20 with 2x 300W panels)
PS. All this is without losses, so you always need a bit more to cover weather, electric losses, etc.

agree with @Richard Mackay that 12V could be simpler - you are basically describing a camping setup. 12V LED's are fairly common and the camping fridge should be 12V (unless it is one of the 220V only types). With the wiring etc. already done though you may not feel like changing circuit breakers etc.
Inlcude a bit more info about your LED's. People could assume they are 5W globes, while they are 200W floodlights...
depending how critical the things are for you that you want to power be careful to not underspec on the storage. If you have 2 or so days of cloudy weather you will be relying on the battery more than you think (if totally off-grid having some spare autonomy is a good idea). If just popping out over a weekend and warm beer/melted butter is not a problem it should not matter that much.
You could potentially even get away with a mobile solar panel setup that you take along if packing space is not an issue (here lithium batteries could be a nice worthwhile expensive luxury)... unattended batteries and solar panels are high on some shopping lists...

If you let me know which laptop your'e using I'll probably be able to provide a DC supply cable so no need for 220V to run the laptop PC.