I need to replace a battery connector on my ESC with the correct style plug. The ESC wire is 18AWG and the replacement
connector I need to put on is 22AWG. Is it safe to solder the thicker wire to the thinner wire of the ESC?

Also, is it safe to plug a 650-750mAh 9.6V NiMH battery pack with a 18AWG wire to the 22AWG wire connection?

ESC's generally have a 3 pin servo style connector, a power connector, and motor leads.
Having said that, the servo connector may be used to connect both a throttle control line from the RX,
and RX operating power from an ESC's internal BEC. The usual current capability is less than 3A max (peak) from the BEC.
The power connector is less standardized, particularly on small models "ready to use" out of the box.
I assume you are trying to use a connector with a pigtail already attached. A 650-750 NMIH pack is not intended to provide high current. They were often used to
power a TX, and usually supplied less than ~100 ma to the TX.
The current should be low enough, although you might notice a small decrease in speed.

If you are using a bullet connector as in an EC3 conne ctor, the bullet will accept a range in wire sizes typically 18-20 gauge wire for that diameter bullet. You should always solder the connector directly to the wire leads of the ESC unless you have to extend the wiring . Going from a thinner wire to a thicker wire (toward the ESC/Motor)is not a good habit to get into. If will work for low amperage loads and short lengths but you should always try and keep the wire size the same or at least go from a thicker wire to a thinner wire (toward the ESC/Motor).
As far as the battery pugging into a small size wire, That is OK.....heavier wire to lighter wire going toward the esc/motor.

The wire nearest to the ESC is the thinner wire. The wire I want to solder to that, going to the battery pack, is the thicker wire.
So it will be the same for the battery pack. The thinner wire nearest to the battery pack going to the thicker wire.