Make sure you've read through all the examples in
the guide, and also into the
Template and
Preview classes documentation.
But basically you can do some simpler stuff. For example you don't really need a full check/true/false block for simple one liners. E.g.:
<a href="/somelink" class="link{{ @is_active ? ' active' : '' }}">{{ @link.name }}</a>
It's also possible to use the following braces format and use (semi-)regular php instructions: E.g.:
{~ if(@something === 'value'): ~}
cool
{~ else: ~}
not cool
{~ endforeach; ~}
Which I personally prefer over the xml tags like check, repeat, etc.
I'm sure I could probably replicate some of your examples but if you're used to Twig, just go ahead and use that. Still the above info may come useful in the future.
As for your other question, I think Base implements array access so you can probably also access it through $f3['twig']->something (not really sure). You can also set it as a class property and use $this->twig->something. Or you could abstract all the initializing and rendering inside some private method, use afterroute, etc.
Good luck.