Is it possible to get significant winds in the Alps if the chart looks like this ?
You can also get Anabatic (upslope) wind during the day
and Katabatic (downslope) wind during the night
Anabatic winds can give you a nice climb rate boost :)
As far as I know austria requires a flight plan if you want to cross D or C airspace, so for Innsbruck you need one.
In some valleys wind direction changes quickly. I did my PPL in Ruhpolding ( not existing anymore, there is a road instead) and the wind changed direction every day between 9 and 9:30 within 5 minutes. Flying through the valleys is not a big deal, but you need to know the area. Some passes are just to high to climb out, so you need plenty of altitude before you enter the valley. Don't underestimate the local winds. Downdrafts are killers, I got caught in one and despite full power I was sinking with more than 1000ft/min. Luckly the valley was wide enough for a turn.
Für Sichtflüge mit zivilen Lfz von Deutschland nach Österreich und umgekehrt ist grundsätzlich kein Flugplan erforderlich. Von dieser Regelung ausgenommen sind Sichtflüge nach Österreich durch kontrollierten Luftraum (C und D) bzw. zu kontrollierten Flugplätzen in Österreich. Für solche Flüge ist ein Flugplan abzugeben (AIP Austria ENR 1.10).
the above cited is a notam for VFR flights to Austria (btw Schengen States). It says that a flightplan is mandatory if you intend to cross controlled airspace ( C and D ) or if you fly to a controlled airfield.
What I can't comment on is flying inside the valleys (what is called "mountain flying") which I have never done and which needs special training.
I have received such training and I think it is grossly overrated. Mountain flying is not difficult or dangerous provided you know a few things.
In the old days, the question always was what is behind a mountain. A wide valley or a solid wall? How wide is the valley, can you safely make U-turns?
Nowadays, this is dead easy. Your GPS shows you everything. If you then know that when there is an updraft, there is also a downdraft and vice versa, you have all you need to do the right thing.
Cross ridges with an angle of 45 degrees so you can turn back quickly. Don't fly in the middle of valleys so you have plenty of space to turn around.
You're pretty unlikely to get Foehn at this time of year, it is a Fall/Spring (and sometimes winter) phenomenon. Btw, if there is Foehn, you simply don't fly a SEP - you've got no chance to outclimb the downdrafts.
Flying to Innsbruck from southern Germany the easy route is of course to simply follow the Inn valley. Can't really miss it! A very scenic route would go through the valleys north of LOWI, then coming down to the Inn valley west of the Nordkette, overflying the Seefeld area.
If you don't know the area, program the route in your GPS, by which I mean a waypoint every few miles and definitely at every intersection of two valleys. From the air - and especially on a hazy summers' day - they all look alike and it is very easy to fly up the wrong one. Always be aware of where you are.
As a general rule for mountain flying, always stay on the downwind side of the valley, i.e. where the wind rides up the ridge. This avoids downdrafts and helps if you need to climb. Fly early in the morning, before the thermals develop, this also avoids the gliders of all kinds, who tend to infest the Alps.
Innsbruck is a very nice town to visit, at this time of year normally overrun by tourists, though. Enjoy the trip!
Ah, one more thing - if you see an N-reg PA28 parked on the apron, say hello. AFAIK she's still there - I did my very first lesson in her!
You're pretty unlikely to get Foehn at this time of year, it is a Fall/Spring (and sometimes winter) phenomenon.
I wouldn't say that, we had some Föhn influence last week. But granted, not strong enough to be a problem for flying.
I have received such training and I think it is grossly overrated. Mountain flying is not difficult or dangerous provided you know a few things.
As an Austrian, about 75 % of my flights (less of my flying time) have been "within" the mountains. I share your opinion when it comes to enroute flying in fair weather. Most things to consider are pretty basic stuff – don't enter a dead end valley when you are not sure that you can outclimb the mountains, stay away from the lee side when winds are strong, better approach ridges at an angle ... That's about it.
What has to be taken very seriously, however, is a/c performance in hot-and-high conditions. An astonishing number of pilots seem to be completely unaware of density altitude and its consequences. Every year, there are several (often fatal) accidents at high-altitude airfields here in Austria like LOWZ, LOSM or LOGM – even though none is a "true" altiport as for instance in France.
Concerning the OP's questions about LOWI, I think most things have been said. If there is Foehn, stay high as long as possible and stay away from the South side (even airliner operations are affected by Foehn). There is a special "VFR departure route in case of Foehn" in the Jeppesen approach plate. You need a flight plan to enter airspace C/D in Austria. Usual routing into LOWI from Germany is via Kufstein (waypoints M1, M2 and so on) or, if arriving from a north-westernly direction, Scharnitz and N1, N2 etc.
If RW26 is in use, you fly directly over the town on final. Quite nice :-).
Enjoy your time there!
Every year, there are several (often fatal) accidents at high-altitude airfields here in Austria like LOWZ, LOSM or LOGM – even though none is a "true" altiport as for instance in France.
In the beginning I was a bit embarrassed about AFIS in Zell am See (LOWZ) giving instructions like "lean your mixture for takeoff" and "reduce your pitch", but there are good reasons for this.
Last time I got these instructions about leaning my mixture on takeoff, I proudly responded that I am going to take off full rich and will show him a 1200fpm climb 
If you do a lot of flying in Austria, a turbocharger very beneficial.
At LOWZ, they have this amazing Cherokee 6 (or Lance) with a really big engine that does scenic flights with 5 Arabs on board at 30C.
Btw, if there is Foehn, you simply don't fly a SEP - you've got no chance to outclimb the downdrafts.
That, of course, depends on the strength of the Foehn. It's an absolutely common wheather phenomen in Alpine regions (unfortunately, because I tend to get headache from it :-) ) and quite rarely really endangers operations.
If it's strong, however - then yes, just stay away.
Last time I got these instructions about leaning my mixture on takeoff, I proudly responded that I am going to take off full rich and will show him a 1200fpm climb
But you didn't use afterburner, did you :-) ?
