@Switch Blayde
The only government "formed" is the President picking his cabinet members and such which are approved by the Senate.
That's exactly what is talked about, only the process is slightly different.
The U.S. president is not all-powerful.
He's not, but there's two roles from ordinary parliamentary system bundled in that figure in US: he serves both as President and as Premier minister who forms the government.
In more run of the mil European Parliamentary system those roles are divided, usually (but not necessarily) leaving almost entirely ceremonial duties to President (although there usually are some mechanisms provided he could use to counter the parliament, was there a conflict) and making Prime Minister explicitly dependent upon maintaining majority in parliament.
"Forming the government" in such a parliamentary system usually goes about this (it may not contain or differ from German specifics):
1) everyone freaks about election results and claims it would be impossible to form a government because no single party has more than a small fraction of the seats (it's common you need three or more parties for majority, and indeed, Dutch have lived "without" government for years (which is misleading statement, in actuality, the previous government is chugging along in those cases, even if without explicit approval of the parliament, or even if formally resigned)).
2) one or more possible coalitions forming more or less stable majority in parliament are formed in talks and backroom negotiations between parties. This typically includes trading ministries and cabinet positions, up the point responsibilities between ministries can be shifted and whole new positions formed just to get a minority partner to commit to vote with the coalition.
3) one or more candidates for Premier Minister are presented to the President. President has formal (but often ceremonial) choice whom he nominates for that role, not necessarily limited by the choices presented.
4) President nominee for the Premier minister forms a cabinet. In theory there's often quite a lot of leeway for them, in practice they're tied by the complex negotiations between parliamentary factions. Cabinet positions are given out to different parties in exchange for support.
5) Parliament votes for approval of the complete Cabinet of Ministers formed by the Premier Minister candidate as a package, complete with Program and Vision (unlike position by position as is US practice). If the vote passes, the new Cabinet is established and goes to work. If it fails, we go back to renewed coalition talks in step (2). This loop can take years, sometimes literally.
6) At any time for any reason or none whatsoever the parliament can vote to withdraw trust from the Cabinet of Ministers. If such a vote pass, we restart the process and go back to the step (2) immediately, although it can be expected that a lot of that heavy lifting is done before the vote of dismissal already.
7) Therefore, minority government is possible, but only as long the parliament cannot even unite enough to dismiss it. It's common to expect Premier Minister to resign if majority in parliament is lost for whatever reason. Successive governments can be approved and dismissed at speed, I think up to six in a year are managed by some countries. On the other hand, highly successful Premier Minister can survive many parliamentary terms and quite radical changes of it's composition as long as his or her party remains part of the majority coalition, even if the Cabinet under them may change beyond recognition in the meantime.
While in US, if President doesn't enjoy majority support in Congress and/or Senate, there's no way around and you have to live in minority government hell.