It's not possible to extend a visa at the airport upon arrival, and in any case the process usually takes days.
It's no problem to tell you want to stay for 60 days.

Having read online job ads for English teachers in Taiwan it seems that a very common requirement is that the applicant is a native English speaker from an English speaking country such as USA, Canada, UK or Australia. I'm not - I only have a Swedish passport.
However, I did most of my Masters education in English and I spent most of my adult life abroad - speaking English both at work and on my spare time. There is no doubt that I'm fluent. Furthermore, I'm planning on getting a TEFL/CELTA certificate. I don't have any experience of formal teaching yet.
It's sad that this is relevant, but I should say I'm male and white.
With all of this in mind, what are my chances at getting a job offer and work visa in Taiwan?
I'd prefer to go on a tourist visa and look for work on location so that I can sample locations and schools before committing. A remote town or village with beautiful nature would be amazing, but any place where I can learn Mandarin is fine. How feasible would that be?
If that sounds like a bad idea, are there any decent recruitment agencies that can help me find a good match?
Thank you for your help!

I'm in Marrakesh right now and I'd like to buy a Tagine pot for a friend, who loves to cook, but I've heard that some of them are display pieces that can't actually be used for cooking.
Is there an easy way to tell whether a Tagine pot is suitable for cooking?
Can anybody recommend a shop in Marrakesh that has a good selection of affordable Tagines?
What's the average price range for a decorated medium size Tagine pot and for plain Tagine pots (the kind with aluminum bottoms and handles that is actually used by restaurants)?
Getting an english teaching work visa to Korea will require a bachelors or at least a TEFL certificate. Ditto Taiwan and Japan.
You might be able to get an informal position somewhere in China or South East Asia(Vietnam, Thailand, Myanmar)
Depending on your nationality, you might be able to get a exchange visa somewhere for a gap year. A lot of schools wouldnt hire you but you might be able to tutor on the side.
depending on what 'ill mental heath' you are talking about, I would advise against staking out into Korea. I taught english in Korea for a year(loved it) but there were a lot of people there who didnt. Many were running from something or thought that a change of scenery would cure all their ills. Most of the time it didnt. Some got trapped and ended up losing a lot of money, or got stuck in abusive relationships with their jobs.

Israel. Tel Aviv, Jerusalem, Dead Sea
It's a hard call, as all of your three options are outstanding destinations in anyone's book.
It depends on your flight times that you will book, as to how much you can squeeze into seeing Israel, ....public transport is very time effective and efficient.
day 1, and (or) day 2, from airport direct to Tel – Aviv. The beach, promenade to Jaffa, bike hire, museums, clubs, cafes, Carmel Market, Neve Tzedek etc.
day 3, transfer to Jerusalem by frequent Egged public bus. Sight see Old City, walk the city walls.
day 4, Jerusalem, early morning start with Abraham Hostel's trip to the Dead Sea, Ein Gedi and Masada. https://abrahamtours.com/tours/masada-ein-gedi-dead-sea-tour/
day 5, sightsee in Jerusalem, Machane Market, Israel Museum etc. and return to the airport with pre booked Nesher shared taxi http://www.neshertours.co.il/en/taxis-from-ben-gurion 64NIS per person. Arrive at Ben Gurion Airport, 3 (THREE) hours before your onward flight.
I don’t know your budget is, but private en suite rooms in either of the two Abraham Hostel, equate to what is known in Israel’s accommodation scene, as good value.
Next time you come to Israel I'll send you to Beersheba, where too many of your daring 4th and 12th Australian Light Horse Brigade are resting.
I hope you enjoy your taste of Israel.