MacBook Pro battery and speed issues

Your problem is (or was) your Power Adapter.

45W that's enough for a MacBook Air but definitely not for a 15" MacBook Pro.

Apple shipped a 60W or 85W power adapter with these MacBook Pro (depending on the processor)

The picture you post might not indicate any problem whatsoever with the hardware. However, the tools and steps to determine if this is hardware or software are quite easy and powerful.

Since the Mac is new, you would be free to use internet or in-person support to ask Apple to help determine if the battery is sound and the software installed properly. You can also check System Information (spotlight will locate it) and check the power settings. If the health isn't Normal initiate hardware service with Apple as software alone won't remedy the low battery life.

To determine if software is the cause, I would do this as follows:

1. Back up your files or the entire drive. It could be hardware and why risk losing anything - whether during troubleshooting or if the Mac fails completely.
2. Make a new user account as administrator
3. Disable automatic log in
4. Shut down your Mac
5. Plug in power and wait for the LED to turn green and disconnect from power
6. Start the Mac and log into your newly created empty admin account

It won't have any saved state, nothing extra running except perhaps any software you have that starts at boot (instead of when your normal account logs in). Run Activity Monitor for 20 minutes and nothing else. Also, document the power in System Information -> Power:

You care about documenting the overall settings and these settings over time as the Mac runs:

  Charge Information:
    Charge Remaining (mAh):     5975
    Full Charge Capacity (mAh): 6432
  Amperage (mA):               -1101
  Voltage (mV):                12096

Open Activity Monitor and go to the Energy tab. Not all the programs listed are running as things from the last 8 hours of non-sleep time are displayed, but you can sort on Energy Impact to show running processes.

In the next 20 minutes where you don't do anything on the Mac except wait for the battery to go down a bit, watch for apps that are using non-zero energy. In general, apps that support App nap and sudden termination will let the OS be most frugal with power when the app isn't busy with work.

After 20 minutes, you should have a good reading on the estimated run time with no load as well as a baseline for what apps are using power before you start loading your menu bar extras, chrome. You can use the Activity Monitor to watch and determine whether your battery can run for 8+ hours idle (it should) and if programs you don't want are causing the CPU and GPU to drain that 8 hour allotment in half the time. Since all shipping MacBook are capable of draining their batteries in 3 hours if the CPU and GPU are loaded 100% - you'll have to determine your energy usage by using Activity Monitor or getting Apple to assist you in testing the battery.

They do have diagnostic tools that can read the power logs and battery information and summarize things more rapidly, so if none of this makes sense you can always get a new Mac checked during the warranty period for proper battery function for free.

This turned out to be longer than I hoped, but my guess is you will find a good battery and that the "light browsing" in chrome is actually one or two tabs that take continual CPU time as well as a few other background tasks that run constantly and prevent the energy savings that depend on App Nap and CPU sleeping for un-noticed to us, but significant portions of the time.