Support

Hello,

I can suggest you to change the collation to your tables, to avoid issues about mixed collations (pro tip: Admin Tools can do that for you).
Please be aware that in some cases you could get a failure during the conversion, since the field is too long for the new collation; I'd strongly suggest you to take a full database backup before continuing.
Regarding the engine, unless you have a site with a lot of traffic, that shouldn't concern you. The main difference between MyISAM and InnoDB is how they actually store the data on the filesystem: the first one uses actual files, the latter writes everything in a reserved area of your filesystem. InnoDB supports transactions, but I doubt that any Joomla extension is actually using them.
On the other hand MyISAM supports the Full Index search index, that could be used by the extensions.

I'd suggest you to change the collation only, and leave the engine as it is.

Admin Tools offers you three options when you change the database collation:

1. Custom (you have to provide your own collation)
2. UTF-8, which is utf8_general_ci
3. UTF-8 multibyte, which is utf8mb4_unicode_ci

Admin Tools will change the collation of everything. It will first try to change the database charset (so you won't have to do this trick again. Usually it fails because you don't have enough permissions).
Then changes the charset of the table and finally of every single column.

If anything fails, it means that it doesn't have enough permissions or an error occurred. Usually it's a problem with the index length, since the index (in UTF8 multibyte) will take more space than allowed to. In that case there's anything we (you and I) can do, since you would have to check if it's possible to shrink the column and then physically change its size.
For the records, Admin Tools will change only the tables of the current Joomla installation.

I'm sorry to hear that, but thank you for the update!

There are a lot of useful and interesting thoughts on the community point of view, but when we're talking about business, at the end of the day it only comes down to the same question: What's the best for me?
Because management is not interested if the tool you're going to use is eco-friendly-locally-developed-gluten-free-zero-emissions, they only care about price and quality: if the perceived quality justifies the price, then they're good.
So the ear we have to talk to is the one about reducing operating costs and easing everyday operations.

First of all, they will state that if is the software is free, how are they still in business?
Let's clear a common misconception: Open Source Software doesn't mean you don't have to pay for that (you just paid for Akeeba software that is open source). That's a linguistic problem coming from the word free, that's why you usually found the statement free as freedom of speak or free as free beer to eliminate any doubts.
Side fact, Linux is free, but the server edition is not free of charge, ask that to the hosting companies...

Then they're going to object about business continuity: I want to go corporate because I know they will be there in the future, I don't trust a bunch of volunteers.
This is a moot point, because we all obey the laws of business. If what you're spending (time, money...) doesn't return enough value back, there's no point in continuing. This happens for corporate (they're in for the money) and for volunteers (they're in for the satisfaction to see your product being used). There's plenty of stories about famous products being discontinued because the holding company was sold or simply went bankrupt (fun website, all the projects Google killed)
However, Open Source Software has a nice thing: the code is completely open. This means that if Company X stops selling it, Company Z can get the code and continue from where it stopped. If you're really desperate, you can keep using the same product and maintain it by yourself. It's terrible, but you can do that if and only if you have access to the source code.

Finally security, because nowadays everything is about security.
Inspecting and debugging closed applications is possible but it's hard. This limits the amount of eyes that could take a look at it and find potential bugs and security threats. On the other hand, when you have the source code, you can run a lot more of tests and inspections and it's way easier to find bugs.

As closing note, for what is worth, the global trend is to move to Open Source. Even the Mother Of All Corporates (also known Microsoft) is moving with an impressive pace in that direction. They started to release their own code (Visual Studio went Open Source), they started to publish debugging info to help inspecting the closed source, they started to contribute to Open Source projects (they submitted a lot of changes to the control version tool named Git); finally they are going to ditch their browser engine to adopt Chromium (Open Source browser engine).

I hope I was able to give you some bullets to spend :)