Summing Amplifier - Resistor at non-inverting input terminal
Could someone please explain to me why do we need a resistor at the non-inverting terminal of the summer amplifier

The resistors R7 and R10 are there to remove a voltage offset at the opamp input terminals due to the 'common mode input bias current.' This is a dc current drawn (equally) by both terminals. The resistor is chosen to have the same value as the equivalent resistance seen by the inverting input terminal (R1-R6 of the summing amplifier, in this example). By doing so, an equivalent current drawn by the input terminals leads to the same voltage drop across the input resistor networks for each terminal, eliminating a voltage offset at the input terminals that would otherwise result.
In many cases where the opamp has very low input bias currents, and where high precision is not required of the circuit, this resistor is not necessary, and the input terminal could be directly connected to ground.

and why do we need an inverting amplifier connected to the summing amplifier?

The output of the summing amplifier has the opposite polarity as the weighted sum of the input voltages. If the rest of the circuit that uses the weighted-sum voltage requires it to have the same polarity as the weighted sum voltage, then the inverting amplifier restores the polarity so that the output is not negated. Sometimes the rest of the circuit could be designed to use the negated voltage instead, in which case it would not be necessary to have the inverting amplifier follow the summer.
A frequent usage scenario would be to use the weighted sum voltage as input to an ADC that only accepts positive voltages, which would motivate using this inverting amplifier so that the output of this sub-circuit has positive polarity for a set of positive input voltages.