Lucene models documents as a collection of fields, each with a textual value.

http://lucene.apache.org/core/4_0_0/core/org/apache/lucene/d...

At search time you can use the default field or specify the fields to be searched:

http://lucene.apache.org/core/2_9_4/queryparsersyntax.html#F...

I can't speak well for Solr, but with elasticsearch the answer depends on what you exactly mean with "search across multiple columns", but it's probably yes.

Yes. Read the example SOLR schema, it's very easy to understand.

Yes you can, and a whole lot more.