It's available and powered by the CirrusSearch (based on ElasticSearch) which is the used search engine in the MediaWiki sites, and the Wikimedia platform.

Some usage examples:

Note: Full guide reference, and source - on Wikipedia: Help:Searching#Search_string_syntax.

As for your question:

Logical operators

The search engine supports boolean logic in searches. The logical operators include the "-" (minus sign) character for "logical not", the AND, the OR, and the grouping parentheses brackets: (_).

Logical OR must be spelled in capital letters; the AND operator is assumed for all terms (separated by spaces), but capital AND is equivalent.

More examples:

• Simple AND, OR: "credit card" OR "payment card", "credit card" AND "payment card" - will work as expected.
• Suffixed tilde character for fuzzy search ("sounds like") - For example, searching for charlie~ parker~ returns Charlie Parker, Charles Palmer, Charley Parks (up to two character difference, not including the first one).
• Wildcard search - searching face*e - starts with face and ends with an e with anything in the middle.

Extra reading

Read more about the search engine and this extension to unveil their full functionality and features.

As far as I know, this is not possible. That's because MediaWiki doesn't actually know what a reply is or that sections on talk pages have some special meaning.

I guess you could write a program that checks all the talk pages you wrote to for replies, but I don't know about anything like that that already exists.

Also, LiquidThreads should solve this (and other problems with Wikipedia talk pages), but I don't think it will be finished any time soon.