Both the circuits you show and any that do the job you require can kill you almost instantly if you are less than competent in what you are doing.
If you are new to this area this is not a good project.
If you are looking to make something useful you can buy it cheaper.
You will never be able to build an inverter of a given wattage from new parts as cheaply as you can buy new commercially made ones. And your results will not be as good if you buid simple self oscillating circuits.
You are apparently in California.
The TIP41/TIP42 transistors are capable of about 6A max and should be run below that in most cases. Pushed to the limit (or beyond) you may be able to get 100 Watt from a pair of them at about 12 volts - but it would not be wise. The circuits shown are aimed at powers in the say 10 Watt to 100 Watt range. Extenbding them to 5A x 110 Vac = 550 Watts out = say 700 wATTS + in is doable but not at all sensible.
At 12V, 100 Watts in current = Power/Volts =~ 8A.
At 700Watts its 7x that or about 60A !!!
A NimH battery will generally not want to be loaded above about 3C - more may be doable but hyas issues. 4Ah x 3 = 12A.
To get 60A you'd be running your batteries at 60/4 = 15C, which they will not happily (or maybe at all) do.
To scope the cost of what you can achieve with those circuits, here are some 100W commercial examples:
Frys's will sell you 100 Watt 12V-110 VAC inverters for $25 here and $20 here with a USB 5V output a s a bonus and $20 here and A 150 Watt one for $32 here and alot more.
If this is for experience, there are a lot of other safer and more rewarding things that you can build cheaper than you can but - unlike these.
Boty the circuits you show will work after a fashion. Both have limitations. The first is essentially pure square wave modified by the transformer inductance. Oscillation frequency is set by R3 R4 C1 C2 and needs playing with and will never be right. A 110-12-12 transformer may suffice but maybe a 110-15-15 due to the square wave and, as the battery will drop from 16+ volts when charged to 12V when flat the Vout will vary by about 25% over the battery voltage range. And more ...
The second design tries to produce a more pure sinewave. Frequency control is better. You would need to play to get output right and voltage change with battery voltage will be an issue.
The "best" modern solution is a 12VDC to about 170VDC inverter and PWM of an H bridge of MOSFETS off that. Vout AC can be controlled by bus voltage variation and/or by PWM CONTROL. Just like you'll find in even quite cheap commercial units in the $20-$30 range. Cheaper designs will use pseudo sinewave with only a few steps - and still probably be netter than what you can easily and cheaply do yourself.