I think pictures would help here…

The rotors got replaced but they didn't do the pads.

You should never replace one and not the other when it comes to pads and rotors. They will not break in properly that way. Unfortunately, mechanics do this at times. I had one do that for me once before I started doing everything myself and it resulted in brake issues that persisted until I replaced both the pads and rotors at the same time.

Unfortunately, they replaced the most expensive of the two, and at some point you will most likely be doing brakes again because of that.

 

I've done pads and not rotors but never rotors and not pads, that's odd. While that sounds normal with were the pad touches, it would be good to have a pic. Here are some pics of my rotors, there is a gap at the bottom. The size will depend on a number of things. The rotors are to dissipate heat so it's not unusual to have some sort of gap to help cool things off.

My Impreza

Kids BRZ, yes I replaced that valve cap

My Miata

 

Is that a sidewall plug?

 

@donkpow No, it's paint. there are two dots on new tires. Doing this from memory, one is where the valve stem should be located, the other is the heaviest part of the tire.

What Do Red and Yellow Dots on Tires Mean? | CHT https://share.google/QM5EDGtQpaRm12WT0

 

Pads can be different sizes/shapes. For example if I remember properly the stock Lexus pads are smaller than the Raybestos pads for the 2017 RX350. So there was a smooth area on the rotor where the stock pads contacted it, and then when I put the bigger Raybestos pads on, the new pads contacted the less-clean boundaries of that smooth area first before contacting the smooth area. But the pads eventually wear to the rotor shape. Yes I tested to confirm that there was no loss of braking effectiveness. At a subsequent brake job I replaced the rotors as well so everything is hunky dory on that car now.

By not touching "an inch on the inside part of the rotor" I assume OP means towards the centre of the rotor. If the new pads that were put on are marked as being for that car, it should be fine.