How do I plan a trip route and transfer it to my garmin nuvi 350?

As I recall the 350 only supports one via point so multiple waypoint routes like you want to use would not work no matter what source you used.

That's correct - no multiple waypoints or preplanned routes on the 350.

You can plan routes with Mapquest - but the Nuvi 350 does not support routes.

What you can do is use a custom POI file and put wayoints in it in place of individual locations...

TripFile
...start point
...waypoint one
...waypoint two...
...desired destination

Then on your 350, you'd put in the destination... then, go back to your TripFile, select the first waypoint - and hit "go" - it will ask if you want to change destinations, or, insert it as a waypoint... which is what you want to do. Once past that first waypoint, do the same for the rest of the waypoints until you get to your destination.

Where/what folder should I save .gpx file to my garmin when it's connected via usb?

You'll need to use the POI Loader to put a custom poi file on your 350.

How does waypoints works? Does it switch to the next one automatically or do I have to manually select it? (Waypoints are in the "wy favorites" on my gps)

You would have to select them manually.

One way to get around the lack of routing on your GPS is to use "Favorites" Just enter the locations you want to visit in "Were to" and save them as a favorite. Then, just select the "favorite" you want to travel to.

As a utility worker, I commonly enter the locations I need to go daily as favorites, then quickly find the closest location I happen to be at.

Works great for me.

I downloaded POI loader, and while in the process of loading POIs (waypoints), it asks me to set "proximity alert". What's is that?

Proximity alerts will warn you through either a tone or sound clip when you approach a point that you have specified for an alert.

There are several FAQs on setting these alerts, read through them for more information on what they are and how to specify them.