If this is because of the Republicans' majority then can we assume Democrats have no power to oppose any nomination or any bill also in the next 4 years?
No. First, Supreme Court nominations and bills require sixty votes in the Senate. So Democrats have direct influence on those, at least until they frustrate Republicans to the point of changing those rules (which has since happened for Supreme Court nominations).
Second, while nominations tend to draw high support from partisans, bills have different political impact. Nominations are procedural. Most voters don't really see the difference one way or the other. The parties are less likely to be united on bills. Some bills won't have universal support from Republicans or universal opposition from Democrats. Particularly among voters as opposed to politicians.
As a practical matter, even if the rules are changed to a simple majority, successful bills will need some Democratic support. Otherwise, Republican politicians reliant on crossover support won't be able to support them.
If all else fails, there is a legislative election in two years (2018 elections; winners take office in January 2019). Not only will that concentrate politicians on things other than pure grandstanding, it is at least possible for the Democrats to win the House or cut down the Republicans' lead in the Senate. And if Democrats can't motivate their supporters to turn out, Republicans could build their leads in the Senate and House.
Democrats have twenty-three of the thirty-three Senate seats up for election in 2018. Plus two independents who caucus with the Democrats. Republicans only have eight Senate seats up for reelection, and only two of them are particularly vulnerable. Dean Heller of Nevada is the only Republican Senator in a state that Hillary Clinton won. Throw in Jeff Flake of Arizona as a vulnerable first-termer from a swing state.
Democrats have ten Senators from states that Trump won. Those ten are going to be looking for ways to convince their voters that they deserve reelection. Republicans are going to need them on some legislative votes. So those ten have influence even if other Democrats don't. Not a huge amount of influence but some.