I wonder why the noun "память" cannot be the subject instead
First of all, it can. Твоя память стала хуже is a perfectly grammatical, albeit just a little bit peculiar way to put this in Russian.
That said, all languages work differently.
In Russian we say идёт дождь (literally, "the rain goes") while in English (and lots of other languages) "to rain" is ...
This is not a shortened adjective here, this is an adverb.
Usually adverbs like this are conveyed using adjectives or "to feel + adv." in English (ему одиноко "he's feeling lonely", ему холодно "he's cold") etc., however English does not have a passive version of "to understand", so it might not indeed be obvious that понятно here is an adverb.
1Do we have here an impersonal, omitted subject?
No. Russian impersonal sentences don't have subjects like "it" in English.
Q2. Is the verb "решать" alone enough to express the idea of "to + infinitive(decide)" in English?
Yes, решать expresses the same idea.
Q3. I'd expected to see the perfective "решить" used instead: in the sense of deciding ...
This is one of the forms of an impersonal sentence formed with the independent infinitive which serves as a predicate and a hidden subject of a pronoun in the dative. The infinitive corresponds to present time, though grammatically it doesn't show tense. If you want to refer the action to the past, you should use "было"
and for future "будет&...
Стать isn't really important here. This is just one particular modality of a catch-all Russian construction that talks about "trouble with" something or other. Which has two fairly self-contained elements, a predicate specifying how bad the trouble is (here, стало хуже), and the "c + noun" part where the noun identifies the source of the ...
Как хорошо! is an impersonal (subjectless) sentence and the role of хорошо is indeed adverbial, it answers the question how? and not which?. But predicates in these sentences do assume neuter gender.
На улице темно - It's dark outside
Было интересно - It was interesting
Вчера похолодало - It got colder yesterday
Проводить время ...
Дано (masuline: дан, feminine: дана, plural: даны) is the neuter singular short passive participle of the perfective verb дать and means "[it] is [not] given/provided".
But Russian Wiktionary also lists it as an "unchangeable predicative" with the meaning "to be destined (to have natural ability) to do something".
"Ему было не дано петь" - "He was not ...
Depending on context "хорошо" may be an adverb and a short adjective.
It's an adverb describing conditions or states:
мне хорошо - I feel good
на улице стало хорошо - it (the weather) became good/nice outside.
It's a short adjective when describing objects and other things:
Вино хорошо - wine is good
Представление было хорошо - the ...
You need to research the topic of impersonal sentences, in which the verb (and sometimes an adjective) (predicate) is always in neuter gender.
когда ей было 22 года is a subordinate clause and so essentially a sentence in its own right and it lacks an agent (subject) being impersonal.
Было is a predicate in this impersonal sentence and thus is neuter.
The reason is that you use negation with genitive.
"слов" is in genitive case and therefore lose the role of subject in the sentence.
If instead you switch to nominative, the verb will change to plural as you expected:
Не найдутся такие слова.
Q4. Are there other expressions that follow this "тебе и решать" word order? Or is this a one-off instance?
This construction, i.e. pronoun in dative + imperfective infinitive verb, may be used with any pronoun as well as with various verbs. For example it occurs in such phrase:
Вы много чего придумали, но мне рисовать == You have suggested a lot of things,...
The noun "память" can be a subject.
Память у тебя стала хуже.
Твоя память стала хуже.
It expresses a fact.
Other variants below.
С памятью у тебя стало хуже.
The same as "У тебя с памятью стало хуже", but somebody wants to accent "память".
Что-то с памятью у тебя стало хуже.
Accents unpleasant surprise, discontent.