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That wire IS the bootstrap (to pull oneself up by the bootstrap). In this case, the upper gate is pulled up by that wire in association with the bootstrap capacitor (C_{BSTA}). The capacitor is “flying” with the source of the upper MOSFET. When the lower MOSFET turns on the bootstrap capacitor charges, When the upper MOSFET turns on the flying capacitor is the source of the for the upper MOSFET gate driver.
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Correct, for long life the breadboard current should be limited. However, IMO, in the name of education and time, it’s OK to push things to the limits.
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You may be interested in this post regarding the MOSFET. It is written from a static perspective but may be a good starting point to answer some of your resistance questions. You will find that the resistance data is in the MOSFET datasheet but requires a bit of detective work. It is also highly dependent on V_{GS}.
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BTW, what microcontroller frequency are you using to drive the bridge controller? This should be in the 10 kHz to 100 kHz range.
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