Cannot get max resolution out of Dell monitor from MacBook Pro Retina (early 2013)
The early 2013 MacBook Pro models only officially supported an external resolution of 1920 x 1080 pixels (via HDMI) or up to 2560 x 1600 pixels (via Thunderbolt).

So, in terms of your first point about your HDMI connection, you're doing better than Apple's official specs.

As far as your second point, it's unclear to me whether you're trying to use the MacBook Pro in clamshell mode or not. I'm going to provide some generic advice that should help, but if not you may need to clarify your question. I'm also assuming your Computer is connected to a power socket and not running off battery.

I would reset both your NVRAM and your SMC. See below for instructions.

Reset your NVRAM

Newer Macs use Non-Volatile Random-Access Memory (NVRAM). It may be worth you resetting this. Here’s how to:


Shut down your machine. Yes, a full shut down, not just logging out.
Press the power button and then press the commandoptionpr keys. You have to make sure you press these keys before the gray screen appears or it won’t work.
Hold those keys down until your Mac reboots again and you here the startup chime.
Let go of the keys and let your Mac reboot normally.


Note: When you log back in you may need to readjust some of your system preferences (e.g. mouse speed, time and date/timezone, etc).

Reset the SMC

Your computer's System Management Controller (SMC) can also be reset. To reset this on your particular MacBook Pro, here’s what you do:


Shut down your computer
Keep the power cable plugged in.
Press at the same time shiftoptioncontrol (on the left side of the keyboard) and the power button
Let go
Turn your computer back on with the power button.


After doing both these, let us know how you go.