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What are some programs/projects that a C++ programmer should do to sharpen his C++ skills and learn new & professional stuff?

Try to write a search engine. A search engine consists of several modules or components that perform various tasks like string matching, graph traversal, interaction with database, etc. This means search engine naturally use many algorithms and data structures, which is ideal as you want to learn as much as possible while doing this project.
 
Let me break down how a search engine in its bare form work:
 
Crawling
A crawler is a program that systematically browses webpages for the purpose of indexing. A crawler starts with a base webpage. As the crawler visits this webpage, it identifies all the hyperlinks in the page and adds them to the list of URLs to visit, called the link queue. Webpages from the link queue are recursively visited, and links and keywords are extracted as it goes. Keywords can be identified by searching for page title, links and headings.


Indexing
Search engine indexing collects, parses, and stores data to facilitate fast and accurate information retrieval. The purpose of indexing is to optimize speed and performance in finding relevant webpage for a search query. Without an index, the search engine would scan every document individually instead of the database, which would require considerable time and computing power.


Searching
A search query is a string that a user enters into a search engine to satisfy his or her information needs. Search queries are often plain text. After a user enters a search query, it is then analysed and matched with the database, and successful matches are displayed out to the user as the search result.
 
Try to implement this all in C++ and you will definitely learn a lot. Implement other features like a web interface (use PHP or Python to communicate with C++ API), add a relevancy system to improve your search results (study Google's PageRank for more on this) and….get creative!

You can develop:

• Banking management system.
• Library management system.
• A program that can convert any integer number to numbers in english, i.e. if you type 562, the output should be "Five hundred and sixty two" .
• Car parking management system.
• A game such as Tic Tac Toe.
• ATM Simulator

Try to built an application for any super market with simple UI using File as database...

Some of the projects that improved my programming skills in C++

• 2D Rendering Engine - Learned a lot about organising class hierarchy and inheritance.
• Protocol Implementation - Serialization/Deserialization, streams and etc.
• Application Specific Memory Manager - Placement new operator, constructor/destructor and etc.
• XML Processing - String manipulation, unicode strings and etc.

You can start with implementing basic data structures like linked list, binary search tree, deque, etc.

When you feel you have enough experience with C++ programming, you can try implementing your own version of smart pointers. (You can read about smart pointers here: What is a smart pointer and when should I use one?) While doing that, make sure you don't have any memory leaks in your program. (You can check this by using valgrind or a similar tool) I did this for a course and I think it was good practice.

Start a project based on something you're interested in and *finish it*. Challenge yourself, there's so much help out there nothing is too hard. For me, that was building an fps game. I decided develop a custom 3D graphics, game, and physics engine and I wanted it to run smoothly on an old iPhone 3GS. It was hard, frustrating, brought me to tears a few times, but was one of the best experiences I've had.

Through it I learned OpenGL and was forced to read and understand documentation I wasn't comfortable with. I now have a great understanding on efficiency in C++. I had to learn and implement linear algebra(the source of my tears) and 3D math for my physics engine/collision engine. Then there were data structures, pointers, properly handling memory, data sorting, art, sound effects, simple enemy AI, pathfinding, server/client architecture, game mechanics, game economy, etc.

Basically I started with my idea which was a bit more than I could handle and and programmed myself into a corner (usually unfixable memory or efficiency issues). Then started over. Each iteration helped me grow as a programmer and correct mistakes, each corner I got stuck in helped me recognize and correct my incompetence.

It's worth it. I decided to launch my game just so I could feel like I accomplished something. Within 3 months it was generating more than I was making working full-time. 4 months later I quit my full-time job:) The scary thing is, I still suck at C++, but I'm getting it and so will you.