Electronic – What type of batteries are these

I'm just going to say that 9V batteries are a bad idea, they generally have a very small amount of power stored in them. You're better off going with a bunch of AA batteries than with 9V batteries, if you're looking to get the best ratio. 9V are only good for the cases where you need the higher voltage of 9V, but have very low current requirements. A small SLA (sealed lead acid) battery is a good choice, I've seen these used before on medium sized robots. They are good for when you need a lot of power to drive motors and such, are very cheap compared to NiMh or LiPo rechargeable batteries, and you don't mind the extra weight. Make sure it's 'sealed', they cost a tiny bit more, and have slightly worse performance, but they are safer and don't require any maintenance.

If space & weight are at a premium, and you don't mind spending lots of $$$, then you can upgrade to LiPo (Lithium Polymer). For example, you could get four of these 3.6V 6Ah LiPo batteries, equivilent of a 14.8V 6Ah battery, and it would weigh less than 1lb, whereas a 12V 7.2Ah SLA battery would weigh around 6lbs. Of course, the LiPo's will run you about $80 versus $20 for the SLA. So as always, there's a tradeoff.

Also, your numbers above are wrong...a standard (huge) car battery is closer to 50 Ah than 5 Ah. But those weigh like 40 lbs, versus the smaller 6lb one I linked above.

I have randomly purchased several of these relatively cheap lithium ion batteries from ebay, originating from China. I have a fairly high quality battery charger/analyzer that I use to test these batteries.

They have all been the advertised voltage, but he capacity is nowhere near what they claim. A 5000 mAh battery tested at 333maH. Another 5000 mAh (claimed 5000 mAh) battery came in at 571 mAh. Clearly these inexpensive batteries are closer to 1/10 the claimed capcity. I have managed to get a refund when I complained to the seller.

I saw a video on YouTube where a person took an 18650 battery bought off of ebay from China (that had a tested capacity significantly lower than claimed) and he took the battery apart. Inside was a minature battery that looked like a triple 'A' battery surrounded by a powder that looked like flour!

In other words you get what you pay for. From my experience, it appears as though batteries manufactured in Japan tend to be of higher quality/capacity, although that can change quickly. If the battery you are looking at has a 'C' and mAh rating, then you might be able to get a refund from ebay if the battery doesn't perform as advertised. Today's lithium batteries appear to be relatively safe as long as you don't use them beyond the abilities or 'C' rating. Fire proof boxes are not necessary for storage, but recommendations from manufacturers still tend to include the instruction to "not leave the battery charging unattended". I suspect that this has to do with the fact that there are a lot of really cheap and poorly made chargers that potentially slightly overcharge the battery (usually through trickle charging which you should not do with lithium ion) which is what leads to fires and explosions.