Take it that the Western Digital Caviar HDD we can see in your screenshot is an external USB device then Boris.
The GPU temp increasing is normal as it is being put under load when you are hooked up to the external display, the bigger the screen the harder the graphics chip needs to work to support the display
that's crap about removing the battery,
Everyone is entitled to opinion but it helps when a person can offer substance to support it, Boris had already said that the original 6 cell battery was worn out so there was no point on elaborating on it, especially when the battery concerned is around 6 years old.
FWIW Boris, below is my canned info for helping folk look after their notebook batteries, both newer and older types.
To prolong the life expectancy of a notebook battery it should be allowed to run out of charge once in a while and then be fully recharged again, this is sometimes called exercising the battery.
It is not a good idea to use a notebook with both the battery and AC adapter/charger in place as it will shorten the life of the battery, this because the battery becomes reliant on being fully charged all of the time and so when the AC adapter/charger is removed the battery will quickly lose it`s charge, not all batteries do have but what many older ones do have is a memory cell (EEPROM) which stores the charge amount/content rating and this is often what becomes affected by continuous charging from the AC adapter/charger, typical symptoms may include not charging at all and only charging to a certain percentage and no further.
To exercise a removable notebook battery you should charge the battery to 100% when you are not going to be using the notebook away from a wall socket power outlet for an extended period of time, turn off and remove the AC adapter/charger and the battery, store the battery in a safe place, reconnect and use the AC adapter to power the notebook while you have access to a mains power socket, every once in a while (1 to 2 months) remove the AC adapter, replace the battery, use the notebook until the battery charge is depleted, connect the AC adapter/charger, charge the battery up to 100% and then repeat the very same steps as above.
Notebook and Netbook users who`s computers have integrated batteries should follow any guidelines provided by the manufacturer regarding best power management and settings but would benefit from once in a while allowing the battery to become empty of all charge by following the above steps, then fully charging the battery back up to 100% and then reverting back to the recommended power management steps that the manufacturer has suggested, these steps may include charging the battery up to 100% and then removing the AC adapter/charger, using the computer until the battery charge depletion level reaches the pre-set minimum allowed, reconnect the AC adapter/charger and then fully charge the battery up to 100% again and repeat the process.
Please note that the above will not help with a battery that has been damaged by being continuously connected to mains power or has reached it`s maximum charge amount limit/life expectancy.
If your notebook battery will not fully charge it suggests that it may be damaged or has been charged the maximum amount of times and so you should not allow it to lose all charge until you have a suitable replacement battery, this because a damaged or naturally expired battery may not begin to charge at all.
Some further information regarding how to prolong the life of a Notebook etc battery which includes calibrating it here