Can a democracy have both representative and direct characteristics at the same time?

Yes - many states in the Middle and Western US are democracies consisting of both elements. On the one hand, they have Legislatures made up of Representatives and Senators - this is, of course, representative democracy since the officials are elected and then go on to represent their electors in the Legislature. Additionally, many of these states have the initiative and referendum process. Voters in states such as South Dakota and Nebraska can gather signatures and place potential laws on the ballot through the initiative process; if the proposed law receives a majority vote, it goes into effect and cannot be vetoed. This is direct democracy since the voters are weighing in directly on the passage of laws, bypassing their representatives.