Are electric field lines always conserved? Suppose we have a positive +q charge and a -6q charge at some separation. Then will every field line originate from the +q and end up to -6q or will there be some extra lines coming to -6q from infinity because of higher charge to get 6 times the number of field lines? That is, will there be any line that does not originate from the positive charge but terminates at the negative one?
I think it should be that every line will originate from the positive and go to the negative, only difference will be in the density of the field lines. Am I right?
Also, if I talk about the flux now, why can I say that the flux near +q will be equal to that near -6q? 
 A: 
Then will every field line originate from the +q and end up to -6q or will there be some extra lines coming to -6q from infinity because of higher charge to get 6 times the number of field lines?

Some field lines will definitely come from infinity.

why can I say that the flux near +q will be equal to that near -6q?

I think you should focus on the flux through any surface enclosing the charges +q and -6q. And they are, according to gauss law, definitely not same.
Flux from surface enclosing -6q will be 6 times more than the flux from surface enclosing +q, even though the nature will be different.
A: 

Then will every field line originate from the +q and end up to -6q 

No, every field line won't end to negative charge.

will there be some extra lines coming to -6q from infinity because of higher charge to get 6 times the number of field lines?

 
Yes, many extra lines will come.

I think it should be that every line will originate from the positive and go to the negative, only difference will be in the density of the field lines. Am I right?

No, you are not right. Only some of field lines will end in -6q from +q.Yes, there will be a change in density gradient around charges.

 why can I say that the flux near +q will be equal to that near -6q?

Flux around two charges would be different. You can simply use Gauss law for flux. 
A: This is really just a footnote to Anubhav's answer so accept his not this one!
Anubhav mentions Gauss's law, and this states:

The net electric flux through any closed surface is equal to $1/\varepsilon$ times the net electric charge within that closed surface.

So if you consider a spherical surface around the $+q$ charge the total flux through this surface will be $+q/\varepsilon$, and similarly the total flux through a similar surface around the $-6q$ charge will be $-6q/\varepsilon$. This tells you immediately that there must be flux lines from the $-6q$ charge that don't end on the $+q$ charge, because for this to happen the total flux through both surfaces would have to be equal.
A: Electric field line start on a positive charge and finish on a negative charge or start/finish at infinity of start/finish at a neutral point.   
Here is a computer generated diagram of electric field lines plotted for a +1 charge on the left and a -3 charge on the right (-6 was not available).

There seem to be a lot of electric field lines which go off the edge some of which never come back and even more which appear from nowhere?  In an ideal world some/most? will in fact come back to a charge.
Two equal +1 charges give this diagram.

As well as electric field lines disappearing into the distance there are two lines between the two charges which terminate in the middle where the is no electric field (neutral point). 
Faraday invented lines of force to explain his observations when he was doing experiments in electricity and magnetism.
He "counted" these lines and according to Faraday there should be three times more lines arriving at the -3 charge then leaving the +1 charge.  This ties in with Gauss and the electric flux (number of lines passing through unit area).
So if electric field lines are conserved then where do the extra lines come from which finish at the negative charge?
Perhaps that does show that electric field lines are not conserved but you you have to remember that that this an ideal situation which tries to show what happens in the real world.
The diagrams of field lines which are drawn are visual aids and do not represent real lines and as such there is nothing wrong with allowing lines to start and finish at infinity. 
One last point is that all the diagrams drawn in this question are not three dimensional which makes counting lines in a 2D representation inappropriate.  
A: In such ideal problems, it is always implicitly assumed that there is no other charges in the Universe. So, if our Universe consists of only these two charges and everything else is neutral, then we cannot assume that field lines start at some infinity and end at individual charges. In fact, lines start from one charge and travel to infinity and then come back to the other charge. This picture is in line with the infinite-range nature of electromagnetic interaction.
Since flux is defined as integrated electric field over a surface, you can enclose your charges with identical (Gaussian) surfaces and attempt to calculate the integral. By Gauss's law, however, this integral is proportional to the enclosed charge. Then, it is trivial to see why bigger charge contains more flux than the smaller charge.
