Having just had a quick look at the Viamare website, i don't think that there is any difference in ferry costs between weekends and midweek (unlike the English Channel crossings) so, from this point of view it doesn't seem to make any difference when you travel. The price lists only show low and high season. I also looked at the cost of taking a car and Minoan seem to be charging about 200 pounds (or was that Euros?) for the round trip for the car itself. There may also be a 30% discount on offer. I doubt that you could hire a car on Corfu for that amount - I usualy reckon of beteen 150 -200 pounds per week during the high season.
Although it is probably not necessary, I would be inclined to book in advance from the UK. This way, you can book the full round trip and may be able to take advantage of any discounts on offer.
I would also be a bit wary of leaving my car in or around a port area for two weeks, unattended - they arn't usually the safest places on the planet! Also, you might incur substantail parking charges which would ofset any possible savings achieved through not taking your car to Corfu.
Check in at European ports in very different from what we are used to in the UK. For a start, yu have to check in at least 2 hours before departure (not the usual 20 minutes as here) as the ships are bigger and will visit more than one port so they have to load the cars in such a way that those for the first port of call (Corfu) arn't blocked in by others going further - it looks like absolute chaos, but it seems to work!
When you arrive at the port you will be directed to a parking area (I've never been to Venice, but at Ancona, you park on the quayside - there is no really defined port area). You must then walk to the ferry company office and check in - take your tickets with you. You will be issued with boardng cards and a windscreen sticker stating your destination port. You then return to the car and wait, and wait, and wait. Ancona has few, if any facilities for buying food etc, so make sure you have something with you, including drinks - Venice may be better, but I don't know.
Quite often when loading, only the driver is allowed in the car - passengers may have to board on foot. This is because, quite often they cram the cars in and there's not enough room for eveyone to get out of the car and also it saves on having hundred of people mixing it with a leathal cobination of trucks and cars!
Disembarkation can be equally chaotic as there are usually none of the announcements we are used to for drivers to return to their vehicles etc. so keep you eyes open and be ready to move as you approach the port.
For accommodation en-route I would suggest you look at the D'Accord Hotels website. This chain covers Ibis, Mercurei, Formule 1 and many others.