How can I avoid downdrafts and icing when crossing the Alps in a small airplane?
Yes, a low powered trainer can fly through tall mountains.
 It is not possible to say how often the flight may be made safely as it depends on specific weather conditions across your route of flight.

This is the same as saying a small sailboat, like a sabot, can sail to the Azores.  It is possible, and might even be safe with proper planning and good weather, but you must take care to plan the trip well every time you make it.

The number one concern I would have flying through the Alps (...or Rocky Mountains, or Sierra Nevadas...) is wind, temperature being a very close second.  A low powered trainer is pretty much at the mercy of the wind much above 5,000' and is absolutely helpless above 10,000'.  A very tame downdraft of only a few hundred feet per minute will exceed the plane's best rate of climb and could effortlessly push you into a ridge or, if you're lucky, a valley or canyon from which you may not be able to climb out.  I use the example of the Cessna 150 in Meteor Crater (elevation only 5,700' in Arizona, USA) of what can happen when an under-powered aircraft runs afoul of nature at high density altitudes.  Ice is not really an issue unless you are willing to fly through visible moisture...so...don't fly though clouds, mist, fog, etc.

I earned many of my hours flying through the Rockies in piston aircraft. The vast majority of my time in the Rockies was flying freight in cabin-class Twin Cessnas with turbocharged, geared, high displacement Continental engines and de-ice kits (a capable airplane).  Even in these planes it was common to encounter downdrafts that far exceeded aircraft performance even when using takeoff power and when lightly loaded.  The turboprops that I flew later on would fare better in that they would hold altitude (maybe even climb) at Vy and max power while carrying a load.  A Cessna 150/152, Tomahawk, or anything else with an O-235 will not do well in breezy mountain conditions.  The large mountain ranges in the US are dotted with light plane wreckage due to unfavorable weather and wind conditions.

There are things you can do to more safely traverse high terrain in a low powered plane.  I learned a few of these the hard way...

1.) Learn proper ridge crossing techniques.  There is a well defined "right way" to approach ridges if you have any doubt as to the wind conditions across the ridge.  I provide FAA documentation only because it is what I am most familiar with.

2.) Set very high weather minimums for trips through mountains.  Twice the standard requirements for VFR may be a good start, perhaps even higher.  Adhere to them during your flight.

3.) Identify alternate airports.  Know the status of the alternates (plowed?), the services provided (fuel?), and their local weather forecasts before you depart.  Take time to note distance and direction to the nearest alternates throughout your flight.  The weather can turn quickly and simply turning around and flying home is typically not an option.  Have frequencies and airport diagrams close at hand should you need an alternate.

4.) Have two realistic emergency landing spots picked out at all times.  You don't need have spots that will allow for a complete rollout, just spots that will permit you to dissipate enough energy to survive the impact. I've built the habit to scan sky, then ground, then sky when flying over rough terrain.

Landing in a field and damaging the plane may be wiser than continuing into worsening conditions.  This is a difficult realization to accept, but attempting to fly through worsening visibility, breezy conditions, and any amount of visible precipitation may kill you while landing in a field may just kill the plane. 

5.) Universal advice, but particularly relevant here...if forced to land, fly all the way through the crash.  Pilots that fly their planes through a crash survive.  Pilots who become passengers do not.

If this sounds like a lot of work, it is! If this sounds overly dramatic, it isn't.  The low powered trainer was not really designed for this type of flight.  You need to take additional measures to make sure that the conditions along the route conform to the aircraft's rather severe performance limitations.
You also need to accept that the plane you describe may be quickly and thoroughly outclassed by mountain weather, weather that even a moderately more capable aircraft (C172-180hp, Piper Archer, ...) would not have a serious problem with.

This flight is doable can be done safely if planned well and flown in favorable conditions.