AWS SES Standard Dedicated IP addresses

How can we make sure that the IPs we receive do not have a bad history? We have seen problems with the first we received.

I believe you can check it by following the method in the following document.
https://docs.aws.amazon.com/ses/latest/dg/snds-metrics-dedicated-ips.html

Can we get 300 IPs at a time? Are there any limitations to the number of IPs we can get from AWS at a time? basically, in the future, our app would require quite a lot of IP addresses to be used for email sending and we need to know would it be possible to request a lot of IP addresses at a time

There does not appear to be a quota on the number of IP addresses.
https://docs.aws.amazon.com/ses/latest/dg/quotas.html

The IP address request form looks like you can apply for multiple IP addresses.
I believe you can apply for as many as 300 at the same time if you have a valid reason, but I am not sure if your application will be approved.