Firstly we, as well as Austria and Sweden, can forget about catching Germany. Two of their remaining games are at home and one of the away fixtures is in Tórshavn against the Faroe Islands. So we have to essentially finish above both Sweden and Austria to make a play-off. We also have a weaker goal difference (+2) than Sweden (+4) and Austria (+10) and having played both of our games against the Faroe Islands already, we will probably have to finish on a higher points total than those two teams to qualify.
The three teams have a fairly similar run-in which will make for a very close conclusion. All three must face Germany (Sweden have the advantage of doing this in Stockholm) and all three have one game against either the Faroe Islands or Kazakhstan. While Sweden did manage a heroic comeback against Germany in Berlin to rescue a 4-4 draw last October and managed to beat the Netherlands 3-2 in their final qualifying game for Euro 2012 in Stockholm, if we are to qualify we must assume that Germany will beat all three and that all three will win their 'easy game'.
So the key ties then are Ireland vs Sweden in September, Austria vs Ireland four days later and Sweden vs Austria in October. Let us start with the latter game. The best possible scenario for Ireland here would be a draw. This would mean we would have to only win one of our two games against Sweden and Austria and not lose the other.
Supposing though, that either Austria or Sweden win - our situation becomes only slightly trickier. We still have to win one and draw one against those two sides - but we would have to win the one against the side who won in Stockholm. Considering Sweden have home advantage against Austria, have more tournament success and are considerably higher ranked, they must be favourites. So if Sweden beat Austria, we must beat Sweden and not lose to Austria.
Unfortunately, the Sweden and Austria game doesn't take place until after we've faced both of them. So we must come out of September with at least four points (preferably with a win over Sweden) and hope that the game in Stockholm either ends in a draw or victory for the side we have already beaten. And that Germany hold up their end of the bargain, sweeping all before them.
Having negotiated that, and assuming we are one of the eight best runners-up out of nine (which would be likely) we would face a play-off. The sides currently ranked second are Greece, Russia, France, Croatia, Hungary, Bulgaria, Albania and England (currently ranked ninth out of nine). Assuming FIFA use rankings again to seed teams, we would then face a play-off against Greece, Russia, France or Croatia. All sides we have lost to under Trap. But keep the chin up and sure we might see you in Rio.