Lawn advice

I’ve attached some pictures of my lawn – I’m a first home buyer and it’s my first time owning a backyard. I’m excited to make it as pristine as possible.

I’ve attached photos of its current state.

1) is this buffalo lawn? 2) how can I restore the dead patches in the corner? I’ve trimmed the plants above it to allow more sunlight to get to the area 3) it looks like i have some patches of weeds filtering in – what should I do To kill the weeds without killing the grass? 4) is there anything else I should know?

It appears like the backyard has poor drainage – so when it rains it gets really really soggy. I’m not sure if this factors into anything