The House of Representatives mistrusts Republicans with the economy, absolutely. They are, perhaps, the worst Congressional politicians the United States has ever seen. It's amazing the damage they've done and the damage the continue wanting to inflict. People like this should not have control over any ecnomy, let alone the world's largest.
The House of Representatives is wholly controlled by the Republican Party. Therefore, it implicitly trusts the GOP with the economy. However, the performance of the House of Representatives is completely lack luster. The laws they have tried to create follow a conservative agenda and do not even pay attention to the economic needs of Americans.
The House of Representative does not mistrust Republicans with the economy at all. Republicans control the House, so that is a non-issue on all levels. Republicans of various political bents, however, perhaps want to persuade the other members to vote like them, and to have similar beliefs. This would create more unity in the house.
No, the House of Representatives trusts Republicans with the economy. The House of Representatives contains Republicans, and they currently have the majority in the House. Of course, any Democrats in the House wouldn't trust the Republican members to handle the economy, but a good portion of the House is Republican and would trust what their fellow members come up with for balancing the budget, etc. Paul Ryan, a Republican, is even the chairman of the House of Representatives Committee on the Budget.
No, they do not think that the republicans can not run the economy and do a good job with it. What the deal is that both parties in politics will not let the other side win, so democrats in the house will not let a good bill pass to help.