Q3.
(1) Last year, Lily 'play' badminton every week (grammatically incorrect)
Take note that also the event has already happened 'last year', it is not simple present. Let's think about another scenario, if this question was asking: *This year*, Lily '_' badminton every week, then Lily is one person, singular term, should be 'plays' instead of play.

(2) Last year, Lily 'plays' badminton every week (grammatically incorrect)
(3) Last year, Lily 'played' badminton every week (grammatically correct)
(4) Last year, Lily 'playing' badminton every week (grammatically incorrect)

Just take note, the 'simple present' was used in 'play'. The event has already happened 'last year', this isn't the simple present. If it was, the verb would need to agree with the subject in number to yield:

(e.g.) Last year, Lily 'plays' badminton every week (grammatically incorrect)

'Play' is the base form of the verbs (the usual grammatical term is the infinitive). These verbs do not need to agree in tense and number. A number of verbs of perception can combine with other verbs in the infinitive in English 'felt', 'move', 'heard', 'screech', as in:

(e.g.) I felt it move.
(e.g.) I heard her screech very loudly.

In these constructions (involving perception), the -ing form is also possible if you want to stress the fact that the actions or events took place in time:

(4) Last year, Lily *was* 'playing' badminton every week (grammatically correct)

Cheers!

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