How to clean the intake filter for the washing mashine

Appliance parts like are generally specific to the manufacturer and possibly model. My usual approach, which has served me well, is:

1. Find the manufacturer's service website.
2. Do a parts search there, based on the model # and part description. Ideally, check the existing part for a label.
3. If that doesn't work, look for a parts diagram of your model. Often these include part IDs expressly for the purpose of ordering replacements.
4. Based on the part number identified in step 2 or 3, search on google and ebay for the part. Usually you can find it from a third-party retailer for significantly less than ordering direct from the manufacturer.

You might also try a local appliance repair parts store -- they may be able to guide you to find the right part if they are familiar with your brand of appliance.

Well... you've got 22 gallons coming out of the washing machine each cycle. You've got about 2 1/2 gallons in the vertical pipe, and another gallon in the horizontal pipe, so you'd need to store another 20 gallons to be reasonably safe if you go down that route. I wouldn't advise it -- drain water has all sorts of junk in it.

Personally, I've never heard of a floor drain that is 2" or less, so your drain should have sufficient capacity to drain the water thrown off by the washer.

So it sounds like there's some back-pressure causing the water not to flow through the drain pipe as fast as it should. I'd start by snaking it, and if that doesn't work, I'd extend the PVC pipe through the hole and fairly far down the drain; this should reduce the pressure drop and force the water through the drain faster.