Trains/car in Italy

You can rent a car at Milano Centrale.

As for whether you will make the 4:10 train, no one can say for sure. If your flight is delayed or if there is a long line at the entry points or baggage claim at MXP, you may not make it. It takes 53 minutes to get to Milano Centrale by train from MXP; trains run every 30-35 minutes, so if you miss one, it may take 90 minutes to get to Milano Centrale. That should leave plenty of time to catch your train, but you only have 75 minutes of wiggle room.

Currently the train you are referring to departs from Milano Centrale at 16:05.
To make that train you need to be on the Malpensa Express train departing from the airport at 14:43. That is 1h and 18 min after your scheduled flight arrival. It is impossible to predict whether that is sufficient for you to deplane, go through passport control (if you come from outside the EU), retrieve your luggage at bag claim, and walk to the station across the terminal, buy the ticket and get aboard the train. Too many variables, starting with the possibility the flight might be delayed. My suggestion is not to buy tickets ahead of time online. Just buy them at the Malpensa train station when you get there. There are more travel solutions later which also require only one train change at Milano Centrale. The last one is departing the airport at 19:43 with a change to a Monterosso bound train at Milano C.le at 21:10 (and arrival at Monterosso 1 minute after Midnight).

Since you are going to Menaggio from Monterosso these are your car rental options.

LA SPEZIA (near the Cinque Terre), then drive directly to Menaggio.
MILANO. There are rental office at/near the Centrale station
COMO. There are rental offices (Avis, Hertz, Europcar) at the Como San Giovanni station.

Como is the closest city to Menaggio and is also along the railway going into Switzerland, therefore, regardless of where you decide to pick up your car, Como is the best place to return it before you get on the train to Luzern.

Do you really need that car for Menaggio? Using it to visit other towns? If you take the train to Varenna (1 hour from Milan, info in Rick's book and at www.trenitalia.it), you can cross the lake to Menaggio by ferry (15 minutes, see also http://www.navigazionelaghi.it/eng/c_orari.asp).

Thanks so much for your help everyone. We are coming in through Zurich, so we will have already been to passport control.
We do need the car for Mennagio because we are renting a villa with 2 other couples outside of town. Also we want to make some day trips from there.
My concern with not booking the train ahead of time, is of course, the train selling out. When I looked at the return schedule from Monterroso, 2 categories had already sold out.

My concern with not booking the train ahead of time, is of course, the train selling out. When I looked at the return schedule from Monterroso, 2 categories had already sold out.

Not true. There is zero chance of a train selling out. The only thing that sells out are the discounted tickets (super-economy and economy), but the tickets at the full fare (called 'Base') will always be available, especially on trains going to/from Monterosso.

I checked both May 17 from Milano C. to Monterosso and May 22 back from Monterosso to Como S. Giovanni and nothing is sold out, there are even super-economy tickets still available. Some of the solutions show the icon of stricken out shopping cart. That doesn't mean that they are sold out, that means that those tickets cannot be purchased online through the Trenitalia website, primarily because those trains are regional or suburban trains not operated by Trenitalia. For example the portion from Malpensa to Milan C.le cannot be purchased through the Trenitalia website (why would you purchase that online anyway?).

To see the true availability of tickets enter:
For May 17:
From MILANO CENTRALE to MONTEROSSO
For May 22:
From MONTEROSSO to COMO S. GIOVANNI.
Note that May 22 is a Sunday, therefore many rental places may be closed except for the airports and maybe train station. Therefore you need to check the rental company website to see the opening hours of the offices on that Sunday at the expected time of arrival. If they are closed, you have two choices. Either you rent the car in La Spezia on Saturday morning, drive to Monterosso and park it there and drive from Monterosso to Menaggio the next day. Or you go to one of Milan airports and rent it there.
For example Europcar at Como S. Giovanni station closes at 12:30pm on Sundays. But Europcar on via Galvani in Milan (near the Central Station) opens all day on Sunday.

In 2014 I arrived at Malpensa from NYC at 1:10 and made it in time for the 4:10 train to Vernazza. I did not pre purchase my train ticket. However, 2nd class was sold out as part of the trip was on a train that required seat reservations. I simply purchased 1st class tickets for 9 euro more.

Roberto and jkc, thanks so much again for your help. I think we will wait until we get to MXP to buy our tickets. Also, after your comments, I am going to check out renting a car right in Monterroso or La Spezio and driving up to Mennagio. Finally dropping it in Como for the train to Lucerne.

If you travel to Menaggio on Sunday, picking up the car in Como may not be an option, as the rental places seem to be closed on Sunday afternoon. Rentals at/near Milano Centrale station seem available all day, I presume with most companies (I only checked Europcar).
There is no rental office in Monterosso. The closest would be in La Spezia. I believe they also might close on Sundays, and maybe even Saturday afternoon, therefore you might have to pick up on Saturday morning, then drive and park in Monterosso for the night.

For driving in Italy, each driver will require the compulsory International Driver's Permit which is used in conjunction with your home D.L. These can be obtained at any AAA office for a small charge, and are valid for one year.

You'll need to be extremely vigilant to avoid the ZTL (limited traffic) areas which exist in many towns in Italy, and of course the speed cameras. If you take the car into Switzerland even on a day trip, you'll also need the compulsory highway tax vignette.

For driving in Italy, each driver will require the compulsory International Driver's Permit which is used in conjunction with your home D.L. These can be obtained at any AAA office for a small charge, and are valid for one year.

You'll also need to be extremely vigilant to avoid the ZTL (limited traffic) areas which exist in many towns in Italy, and of course the speed cameras. If you take the car into Switzerland even on a day trip, you'll also need the compulsory highway tax vignette.