My company is providing me 7 days invitation letter for Schengen visa but I want to extend my trip to 20 days, and I am planning to couch-surf or look for BNB while I am there and avoid pre-bookings. How can I have a combination of work and tourist Schengen visa? I am traveling from India and my country of entry in Europe would be Lithuania.
There are several misconceptions we need to clear up here.
First, what you need is not a work visa. There are two possibilities:
It is very likely that, in the jargon applicable here, your planned trip is not "work" at all, but "business". If you're being paid for going on the trip as part of your employment outside EU/EEA, and you're there to participate in meetings, training or the like, for planning/enabling/supporting work you're carrying out back in your home country, then it will generally count as a "business" trip.
On the other hand if it actually is "work" (that is, you're being paid by a local business, directly producing value while you're in Europe, or something else that does not fit the idea of a business visit), then you need to be allowed to do so by the member state where you do the work, according to its own national rules. The rules for whether outside nationals may work are not harmonized at the EU level, so you will need both a work permit of some form and a short-stay Schengen visa. You would then use the work permit as supporting documentation for the purpose of your trip when you apply for the visa.
Second, there is only one kind of short-stay Schengen visa. Once you have the visa, you can enter on a trip for any legal purpose, whether that is tourism, business, work (to the extent you can legally work, which, again, the visa does not determine), or a combination of these.
When you apply for a visa, you will document the particular plans you have for the trip. If that is a combination of tourism and business, provide a combination of documentation for tourism and business. You can tick several of the boxes in the "purpose of trip" field on the visa application.
Third, it is not particularly relevant what your "country of entry in Europe" is. What is relevant is where the main destination of your trip is -- in your case that would be the location of the company you're visiting. If that is Lithuania, then your visa application must be processed by Lithuania no matter where you will physically cross the Schengen external border.
Finally, a tourist visit without prebooked accommodation does not make a very strong visa application premise. It you have a good strong premise for the business part of your trip, it is possible that the consulate will be somewhat relaxed about our incidental tourism plans, but you can't really rely on that. Your chances of success will probably improve considerably if you can present more definite plans than this.