Olive oil can't take high heat. The expensive extra virgin olive oil is more heat sensitive than the cheap olive oil.  I do however cook eggs using olive oil because eggs are cooked on low heat.

Green tea is acceptable probably not paleo.

Olive oil is not technically paleo as it requires a certain amount of processing.  However, it can be difficult to find truly paleo fats, so some of us use things like olive oil and grass finished butter.

Chickpeas are legumes and definitely not paleo.

Sugar free gum is not paleo but I doubt it's all that harmful.

Beetroot, being a root vegetable, is generally considered paleo.  Note that tubers like sweet potatoes are not.

I actually consider green tea paleo since it's just dried leaves.  Even if it's not, it's likely not very harmful, as long as you don't add anything like milk or sugar.

Well, in regards to olive oil and your paleo diet recipes, I would say it has more to do with fat content. You want to use oils as little as possible because all oils are is concentrated fat, empty calories. In fact, I would be careful how many olives you eat because they are brined and contain a lot of sodium. I've use olive for cooking all the time. Just don't over heat it or it will change the flavor of it.

If you're scared of eating fat, you're in the wrong place!

Here is a good tool for oils
http://www.eatingrules.com/2012/02/cooking-oil-comparison-chart/
higher is better, farther right is more heat resistant. Most here us coconut, tallow, or grass fed butter.

I'm not sure how much I agree with that chart.  Hazlenut, macadamia, avocado oils etc. are OK for salads but I wouldn't cook with them, when there are many better alternatives!

Fats for cooking
- grass-fed lamb and beef tallow is the 'king' of fats but as it's high in sat fat I sometimes find that it congeals a little unpleasantly, although this is more of an issue with lamb fat
- grass-fed ghee is OK fat-wise, and delicious.  I like it because it's saturated enough not to oxidise very much during heating, but not so saturated that it will solidify on food as it cools.  However there are some concerns about hormones contained therein
- lard is my personal favourite for cooking in terms of flavour and consistently but it is very, very hard to find to pigs that haven't been fed soy and/or grains
- chicken and duck fat is last in terms of animal fats
- coconut oil is by far the best vegetable fat for cooking and in my experience pretty flavourless

I agree with celticcavegirl.  That chart assumes saturated fats are bad and monounsaturates are good, which doesn't seem to be the truth.  My experience before and after paleo was also that tallow is better at high heat than refined soybean oil.  Random things on the other side of the kitchen collect a lot less grease now that I cook with more saturated fats, and the stove and cooking implements are much easier to clean, so it seems like the animal fats throw a lot less of that stuff in the air.

I've been in a lot of restaurant kitchens in my previous job in fire and safety.  Many of them have more grease than Jiffy Lube. They are just caked with grease.  Watch an episode of Restaurant Impossible on Food Network. Many places you eat look like some of the restaurants featured on that show. Of course many of them use a liquid oil called Grill and Pan Oil distributed in 35 pound boxes.