[D]o I understand correctly that the elected Representatives from these states should be disqualified if the election in the states will be ruled "unlawful"?
No, the relief requested by Texas is limited to the electoral college. That being the case, the Court would grant no more relief than requested. It follows that the elections for US Representatives and Senators (and for state and local offices, etc.) would not be affected.
MOTION FOR EXPEDITED CONSIDERATION ...
Texas also will move for interim relief in the form of a temporary restraining order, preliminary injunction, stay, and administrative stay to enjoin Defendant States from certifying their presidential electors or having them vote in the electoral college.
In a contemporaneous filing, Texas asks this Court to vacate or enjoin—either permanently, preliminarily, or administratively—Defendant States from certifying their electors and participating in the electoral college vote. As permanent relief, Texas asks this Court to remand the allocation of electors to the legislatures of Defendant States pursuant to the statutory and constitutional backstop for this scenario: “Whenever any State has held an election for the purpose of choosing electors, and has failed to make a choice on the day prescribed by law, the electors may be appointed on a subsequent day in such a manner as the legislature of such State may direct.” 3 U.S.C. § 2 (emphasis added); U.S. CONST. art. II, § 1, cl. 2.
Concerning the contingent election
It depends on whether an electoral college vote takes place in the affected states and whether the results are certified before January 6, 2021. The proposed schedule for the case extends to late December 2020. Though other scenarios are possible.
Whether there is a contingent election in the House actually depends on whether no candidate receives a majority of electoral votes after any challenges to the states' electoral votes are resolved. If a candidate receives a majority after challenges there is no contingent election. See, Amendment XII.
In any case, the representatives of states with challenged slates (one or more) of electors will still participate in votes as to whether those slates of electors will be counted. Challenges to the votes of electors, during the counting of the votes, are permitted by 3 U.S. Code § 15 - Counting electoral votes in Congress.
In deciding whether to reject a state's vote, the House and Senate vote separately and, if both agree, the votes are not counted. Voting in the House is by all representative, not by state.
It follows that, if all Democrats in the newly-formed House accept all electoral votes for Biden and Harris, then Biden and Harris will have majorities of the electoral votes and will become president and vice-president, respectively. Thus the only real possibility for Trump and Pence to win is to prevent the challenged states from submitting electoral votes for Biden and Harris on December 14th.