Of course they were a third party. The fact that they gained prominence quickly does not change that.
A third party would mean they where a challenge to the two party system. In their first national election, they where one of the two main choices, they never where a ‘third option’. The Whigs had already been pushed out. 1860 was a two party race.
No, they weren't.
Between the election of 1852 and the election of 1854 the Whig party ceased to exist and the GOP formed.
In the 33rd Congress there were 77 Whigs and no Republicans - the GOP had not yet formed.
In the 34th Congress there were no Whigs and no Republicans. The Republicans formed as a result of this election.
In the 35th Congress there were 90 Republicans and no Whigs, in the very first national election that the Republicans contested.
The GOP started life as America's second party, not its third.