1-- The answer should be yes. ' socialising' means talking with your friends, maybe going to a party, or even looking for a romantic partner. It is a casual, relaxed setting, whereas as work is work and is more formal. You shouldn't, for example, use swear words or tell jokes at work.
2 -- The sentence is entirely correct. I might help to see that 'My grandma' could be exchanged for 'she' :
"She remembers washing her clothes... " ('remembers' is the main verb conjugated for the subject 'My grandmother'--- 'washing' is the verb "to wash" in the form of a gerund, which is always the present continuous form of the verb)