What are /r/TYPO3's favorite Products & Services?

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It depends on how you installed your site. Is it composer installation or did you download the files from the typo3.org? If composer based - did you use helhum's secure installation?

But most importantly - why would you want to access index.php? Do you want to change the template? If so - you are not looking for index.php, you are looking for a template defined in sitepackage. Look inside sitepackage/Resources/Private/Templates...

I suppose it's too late for the OP to use the recipe, but I think Heroku's free tier might work.
Things to consider:
1) As a free dyno gets idle on inactivity, you will need to set up a monitoring system (for example, I used free https://hetrixtools.com/ on a couple of projects) to make requests and keep dyno awake.
2) The filesystem is limited to 512 MB, so if your site is big, it might be not for you. Or you can offload everything to S3.

PS: Overall, the hosting is typically cheap comparing to the cost of labor. So I am surprised that TYPO3 developers have such issues.

You shouldn't need to. The export module should take care of that as well AFAIK. But the best place to get help is probably slack - https://typo3.org/community/meet/how-to-use-slack-in-the-typo3-community/

Well, a good start might be the TYPO3 books, like this one: https://typo3.org/news/article/book-modern-extension-development-for-typo3-cms-with-extbase-fluid/

This should get you covered on all the recently developed extensions as well as most of the modern core (version 6.2). Also good reading is the frontend rendering process explained here: https://buzz.typo3.org/uploads/media/TYPO3_Frontend_Rendering_Process_v1.5.pdf

However I have been mostly developing TYPO3 websites, so I do not know what kind of things you are referring to. (Optimalization or refactoring or adding small features?)