This is fine. Visitors to the US are classified either as visitors for business, B-1, or for pleasure, B-2. (The analogous status codes for those who have entered under the Visa Waiver Program are WB and WT.) Although there is a dual-purpose visa, the B-1/B-2 visa, there is no combined B-1/B-2 status. Because of this, you should be admitted to the US in B-1 status.
Nonimmigrants who have entered the US for one purpose may generally apply to change their nonimmigrant status for another purpose. It is not necessary for a business visitor to do this, however, when combining pleasure travel with a business visit. The US page on change of nonimmigrant status explains:
You do not need to apply to change your nonimmigrant status if you were admitted into the United States for business reasons (B-1 visa category ) and you wish to remain in the United States for pleasure before your authorized stay expires.
Business travelers frequently use their paid business travel as a springboard to personal travel on their own expense. This is incredibly common, and is highly unlikely to cause any concern at your immigration interview.
I'll be on tourism & business purpose even though my letter clearly says I'll be visiting Texas for the summit(Business) for approximately 2 weeks.
The letter serves as evidence of the main purpose of your trip; it shows what you'll be doing during the first two weeks. There's no reason to hide the fact that you're planning to stay longer in the US (indeed, the officer is most likely looking at your airline reservation while he is interviewing you). If the officer asks, just explain your plans. It's all perfectly legitimate.
Furthermore, invitation letters are not generally required to enter the US. They are useful to support your assertions about your travel plans (for example, that you're attending a summit in Texas at your employer's expense), but it is certainly not necessary that they describe details of your itinerary that do not concern your employer.
Will I be questioned about why am I traveling back from NY or to show my domestic flight from Austin to NY ?
It's possible, but if this happens there is no cause for worry. It is also possible that the officer might want other evidence relating to the latter portion of your trip. For example, he may want to know more about your friend, or to speak with your friend on the phone to confirm that he is expecting your visit. This is, however, very unlikely unless the officer becomes suspicious that you are lying about something or otherwise planning to do something improper.
To reiterate, mixing business and pleasure in a single visit is perfectly normal, and having an invitation letter that mentions only the business travel in your itinerary is not a problem.