Сегодня мне пришла квитанция за коммунальные услуги, и я сразу проверил счётчик воды.

Мне пришла квитанция uses a common Russian pattern where the “receiver” is in the dative (мне) and the thing that arrived is the subject (квитанция). Literally it’s like “To me came a bill/receipt.”
You can also say я получил(а) квитанцию, but it’s a bit more “active” (“I received…”). The мне пришла… structure is very natural for mail/notifications arriving.

Past tense verbs in Russian agree in gender and number with the subject. The subject here is квитанция, which is feminine, so you get пришла (fem.).
If it were письмо (neuter), you’d say мне пришло письмо; if счёт (masc.), мне пришёл счёт.

Мне is dative singular of я. The dative often marks an indirect recipient/beneficiary, especially with verbs like приходить (to come/arrive), звонить (to call), нравиться (to be pleasing), etc.
So мне пришла квитанция = “a receipt came to me / for me (as recipient).”

Here за means for (payment for / in exchange for). With this meaning, за typically takes the accusative: за коммунальные услуги.

In Russian, many plural nouns have the same form in nominative plural and accusative plural, especially for inanimate nouns (and услуги are inanimate).
You know it’s accusative here because the preposition за (meaning “for payment”) requires the accusative, even if the form doesn’t visibly change.

Because it joins two independent clauses (each has its own verb):
1) Сегодня мне пришла квитанция…
2) …и я сразу проверил…
In Russian, you usually put a comma before и in this situation (compound sentence).

Yes, it can often be omitted: …и сразу проверил счётчик воды.
Russian verb endings already show person/number in the present/future, and in the past they show gender/number, so pronouns are often optional. Я is kept here for clarity/emphasis and to make the subject switch explicit after the first clause (which started with мне, not я).

Проверил is perfective: it presents the check as a single completed action (“I checked it (once), done”). That fits with сразу (“immediately”).
Проверял would be imperfective, suggesting a process, repetition, or background action (“I was checking / used to check / checked for a while”), which doesn’t match as neatly with a one-time immediate reaction.

That’s the standard past tense marker. In the past, Russian verbs agree in gender/number:

• masculine: проверил
• feminine: проверила
• neuter: проверило (rare for people)
• plural: проверили

So a female speaker would say: …и я сразу проверила счётчик воды.

Воды is genitive singular of вода. This is the common “noun + genitive” pattern meaning “X of Y / Y-type X”:
счётчик воды = “(a) water meter” (literally “meter of water”).
Similarly: стакан воды (a glass of water), бутылка воды (a bottle of water).

In this context счётчик is a “meter” (like a utility meter: water/electricity).
The correct spelling is счётчик with ё. In many texts ё is often written as е (счетчик), but the pronunciation remains [shchyot-]. Seeing е instead of ё is normal in everyday writing.

Word order is flexible, but different choices change emphasis:

• Сегодня мне пришла квитанция… (sets time first; very natural)
• Мне сегодня пришла квитанция… (emphasizes recipient a bit more)
• …и я сразу проверил… (emphasizes immediacy)
• …и я проверил счётчик воды сразу. (also possible, slightly heavier ending emphasis on сразу)