There is a hotel guest who has lost his wallet and he goes to the cleaning lady and asks if anyone has turned it in. She asks if he has checked with the woman in charge of the floor. He responds, Да, мы с ней вместе всюду искали. It seems the action of "searching" is past and finished, not repeated. Why is it in the imperfective aspect?
They spent some time serching for the wallet and didn't find it.

Russian Verb Aspects - Russian Grammar

8. If there is no result of the action, or it wasn’t successfully completed then use the imperfective aspect.


In another section, the cleaning lady is advising the guest to avoid meeting certain suspicious people in the city. She says ...вам всё равно лучше избегать встреч с такими людьми. I would expect an infinitive after избегать. What is встреч in this sentence?


It is a noun (feminine, plural, genetive).