Another thing that I have hard time understanding is how if I hook the hose to drain in the sink would be different than if I make a direct connection to the drain pipe of the sink. It is still the same drain pipes.
This is the crux of the issue. Your washer ejects a certain gallon per minute, and to ensure that all the water goes down the drain safely instead of backing up and overflowing all over the interior of your house it requires a pipe with a minimum diameter to ensure sufficient water flow.
Which you don't have for the washer you now own.
Draining the washer into the sink can work because as others have pointed out, the sink acts as a buffer - the water that can't get down the drain fast enough can back up into the sink (instead of the floor) thus giving the drain sufficient time to drain the water out. Unfortunately for you, your sink isn't big enough to provide a sufficient amount of buffer. Even if your sink was larger, I doubt code would allow a washer to drain into a kitchen sink for sanitary reasons as others brought up.
That's pretty much it.
The washer you have has too powerful a drain pump for the size pipes you have and your location doesn't offer mitigation such as draining into a sink.
The other washer can get away with a smaller diameter pipe because it's pump isn't nearly as vigorous as the pump your current washer has, so a smaller diameter pipe can accommodate the flow. However, just because the manufacturer states it will work, if local building codes have a minimum size pipe for a washer (which it sounds like is the case for your city) you still won't be able to do it.
Sometimes building codes can be a pain, but for the most part they are excellent and keep people from doing really stupid stuff. Especially in an apartment where a flood of water will affect more than just you - your building owner will be out as well as potentially your neighbors too.
If you know someone who is good in making cabinets it might be easier to put the washer in a cabinet on oversized rollers to make it easy to move and roll it into your bathroom. The bathtub should be able to handle the draining. For water supply I have the add on shower heads with the hoses - I take the shower head off the end of the hose and I found outdoor sprinkler riser pipe is the same diameter and thread pitch, so I screw a segment on the end of the hose and it works perfect for filling my humidifier bottles as well as buckets for my fish tanks. You could use such a hose to create an attachment for the washer to fill from. You wouldn't be making permanent changes to the apartment. As long as you don't forget to leave the drain in the bathtub stopped up, you shouldn't have any issues with discharge either.
The only potential downside is on wheels the washer may vibrate/walk too much when spinning. And front loaders tend to be heavy with the internal counterweights to help with vibration during the spin cycle so it may not be practical to roll either. Finally, you have to store it somewhere when not in use.
Quite frankly I think it would probably be a lot easier to find an apartment set up to accommodate a washing machine. Or see if you can convince your landlord to split the difference on retrofitting to properly support a washer (if it's even possible). You can point out it will make the unit more attractive to future renters. Of course he may realize for that reason you are now not paying enough rent too, so it can cut both ways
