How can the direction of the electric field of a negative point charge be going in towards itself (radially), when the charge itself is generating the electric field? Is it generated at infinity then? Because if it's generated by the charge then it cannot go further in towards itself?
In case of positive charge, it is understandable that the field is being generated at the charge (at it's surface to be precise) and is going outwards radially.
There is no "going" going on in field-line diagrams. The direction of the field lines indicates, by convention, the direction of the electrostatic force experienced by a positive test charge at that location.
Field lines do not indicate the 'flow' of any physical quantity, and there is nothing being 'generated'; instead, all you have is a force field, and ways to study and analyze it. This extends to the concept of electric flux (i.e. for a given surface $S$, the integral $\iint_S\mathbf E\cdot\mathrm d\mathbf S$): we call it 'flux' by analogy, but there's nothing at all actually 'flowing'; instead, it is just one more tool to understand and analyze the force field and the laws that govern it.
For more on field lines, see Why does the density of electric field lines make sense, if there is a field line through every point?.
Firstly, the fields lines are by convention inward for a negative charge, and outwards for a positive charge.
Secondly, fields lines don't actually exist; they are just a fictitious entity that give us as physicists a way to visualize the situation.
Lastly, you will need at least 2 charges to be able to 'view' the field lines. Analogous to Newton's law of Gravitation; you need at least 2 masses to observe the effect of the force field on a 'test mass'.
Note that this does not mean that the gravitational field or electric field is already present in the absence of a test mass or charge.
The direction of the electric field always indicates that a positive charge, located at any point in that field, would feel a force in the direction of the field.
In your example, the negative charge is still the source of the field and there is nothing at infinity in your problem; a positive charge placed in the field will be attracted to the negative charge, hence the direction of the field lines towards the negative source charge.
It's merely a convention. The thing you have there is something called a source charge. To test the effects of a source charge we bring something called the test charge (Note that electric field can be measured only through the force on the test charge, given that we know the explicit charge on the test charge.) A test charge is positive by convention.
It would be handy to familiarise yourself with the definition of electric field. Electric field at a point is the force that a unit positive charge would experience if placed at that point. When we give a visual of this electric field, we actually draw lines of force (a 'line of force' simply tells where the test charge would go if placed at that point; where the test charge goes is dictated by where the arrow points).
PS: 'Lines of force' is a really confusing term. Most books prefer 'field lines' because in the case of magnetism, the field lines don't actually point to where the force takes a charge.
If electric field is the force on per unit charge +ve or -ve then its direction would be the direction of force.
