I understand that your method of electing a president funnels you into a two-party system.
Where I live, anybody can start a new party, collect supporters (usually by having people sign up in the streets) and present a certain number of written endorsements. It's not exactly like that, dunno the exact lingo, but if enough people will put their name on "I think this party might be a good idea"-paper, then that person can run for the danish parlament. The backers don't have to vote for this party or pay a fee to this party. They just have to say "I think this party should be allowed to be on the ballot for the next election".
Then that party must acheive at least 2% of the votes to actually get a seat (two seats, since there are 179 seats in total). This system ensures there's a wide number of parties to represent the citizens. After the election, the Queen appoints a Royal Negotiator, usually the foreperson of the largest party, who then attempts to form a government with 90 members of parliament behind it. By negotiating with other parties, promising ministery posts and changes to laws in exchange for votes in the parliament, the Royal Negotiator hopefully reaches a solution which can then be presented to the Queen. The Prime Minister usually is the foreperson of the largest party in the coalition of parties working together.
That way, small parties with only a few seats in the parliament can effect significant influence, by "selling" its votes to the coalition which offers the most in return.
My understanding is that whether a political system veers into a two-party system or a multi-party system depends on whether it has single-member electorates or multi-member electorates.
Britain can take credit for introducing single-member electorates, and this system was copied in the USA, Canada and Australia among other countries. All these countries have largely gone to a two-party system (yes, there are exceptions, but they are usually based around very specific ethnic issues).
Countries which have introduced some form of multi-member electoral system - most of Europe I understand - have a broad range of political parties.
The difference is that the target number of votes required is much more concentrated in single-member electorates, and this is hard for a party with broad but diffuse support to achieve. Here in Australia the Greens generally get a higher proportion of the vote than the Nationals (junior coalition partner to the Liberals), but the Nationals' support is concentrated in rural electorates while the Greens' support is spread across the whole country. The result is that the Nationals have 10 seats in the House of Representatives while the Greens have only 1.
However, in Australia (unlike in the UK and Canada) our Senate is elected, with equal numbers of senators from each state. But while that might seem to make Australia like the USA, we have 12 senators per state (and two per territory), with six seats up for grabs each election. But in the USA only one Senate seat is ever up for grabs at each election, meaning that Senate elections in the USA are like Reps elections, meaning the two-party system applies in the Senate too. But in Australia we have a proportional representation voting system for the Senate, and this means that minor parties can win Senate seats with relative ease.
So our Senate has a cross bench of 14 out of 76 Senate seats (including 9 Greens). This means the government of the day usually has to work to pass legislation, either by enlisting Opposition support or assembling some one-off coalition of cross-benchers. On the one hand we have the political stability of a government which holds a majority in the House of Representatives, and on the other hand minor parties are able to influence legislation as it works its way through the Senate so the government doesn't have it all its own way.
