First of all they have all kinds of legislation, last time I looked there were over 200 bills, that McConnell refuses to put to a vote in the Senate. Many of them are bipartisan too. McConnell's logic is that Trump won't sign-off on them. He is not acting like the House is a co-equal branch of government. The claim that the Democrats don't do anything is false though because of what I just said. Also, they weren't elected only to legislate, they were also elected to put a check on Trump.
Are you mixing up a vote for an impeachment inquiry - which is not required by the constitution, with a vote to impeach? There will be a vote to impeach. I don't buy the logic that it will be bad for them November. It could be, but I think it is the right thing to do. It was bad for the Republicans in the short run when they impeached Clinton on obstruction of justice and for lying (before Congress) about an affair. Trump has 10 cases of obstruction in the Mueller report, I don't know why that wasn't taken seriously, it was obstruction of a much more serious matter. People disapproved of the Clinton impeachment, but there weren't enough votes to remove him in the Senate. Except among Trump's base, I don't think it will be unpopular to impeach him.
There will probably NOT be enough votes in the Senate to remove him either, but as I said, it's not beyond the realm of possibility. At some points Republicans may decide that he is too much of a liability.