You play it by putting it in and turning the power on. It should be a normal cart, no finger wizardry or special incantations needed.
That it is not working now speaks to some form of hardware trouble, which you will probably want to fix before dumping.
As well as more exotic forms of damage (we have seen things go for a swim in salt water, be left in a damp shed and more besides) the three usual suspects were mentioned above and all could lead to the symptoms you describe.
Cleaning pins is easy enough and that it worked differently on second insertion means even if this was not the usual failure such things see then more likely to be this. Contact cleaner (don't use automotive cleaner, get an electronics one), pencil eraser (go with the pins rather than across them), tile cleaner pen (find a fibreglass one which is what more expensive electronics options will use) or something that works to make metal shiny but will leave the other things alone (if you have some fine sandpaper then be very careful but it too will get it done)... many options for this.
Bad battery has two elements. 1 if the device is not designed to work without it (oh no can't find date and time, better crash) and 2 if the battery leaked and messed up some traces inside the device (not so common with things of this vintage as much as 5 years earlier but could be the case).
Bad chip joints is commonly seen in commercial games*. It is in some ways a design failure of the board itself (if you put force on a joint made of solder, and GBC boards you can see in the video below are designed such that the force experienced might well be as large as it can be, then it will experience something called creep over a longer time period even if it is not enough to snap outright). I don't know what this board will look like inside as it is likely quite custom but at the same time it was a cheap device (remember seeing them in high street game shops in the 90s for £20 or so when games were £10 to £15 more) made by whatever electronics company was willing to risk a call from Nintendo's lawyers so I doubt quality was their primary concern and the same failure modes will be in play in this as well.
*choice video
Unlike a lot of electronics fixing then this will not see you have to know much of anything other than how to heat the joint up and maybe apply a tiny bit more solder, which is realistically anybody that can solder.
After that, or maybe showing us a decent picture of both sides of the PCB, then we can look for more interesting failure options like broken traces and burned out components.
Dumping it is possible, various devices and methods exist for this but if it is a GBA game most would probably just find a DS or DS lite and a DS flash cart to dump it with DS homebrew. I doubt it is anything exotic to dump like some multicarts and very rare examples in commercial games.
If it is a GB/GBC game (search says GBA most likely for the names you gave) then that is a bit harder as GB/GBC dumpers are similarly rare these days. Don't know what we have on the commercial side of things other than
https://bennvenn.myshopify.com/products/usb-gb-c-cart-dumper-the-joey-jr
Playing it, as mentioned, is harder as not many flash carts bothered to support it and even then it might be some extra hardware above and beyond what is seen in other devices -- the GB, GBC and GBA did not feature an internal clock so any available in games (and such a thing would be a requirement for a personal organiser) were extra hardware in the cartridge itself.