Urgh, I work in HR and find this sort of thing a bit cringey (have colleagues that love it though!). The thing is that as much as HR people imply that 'company culture' is this fixed entity that you can decide what/how you want to be and can change at the drop of a hat just by putting on some events or putting up posters or whatever, in reality it's much more nebulous, an organisation's culture is made up of the attitudes, beliefs and behaviours of the individuals in it and how they relate to one another. It tends to be heavily influenced by the history of the organisation, its wider context and norms in the industry/sector and crucially how it is led and managed. With no offense to you OP, I'm sure you do a great job, but it's ridiculous for your boss to expect you to be able to simply change deep rooted and long standing cultural behaviours/issues by getting people to put on silly hats on zoom
That being said, there's a lot to be said for approaching the issue of culture from 2 angles, one is setting basic expectations and ground rules and then enforcing these and the other is encouraging the kind of behaviours and interactions you want. For the former it's common for organisations of all types and sizes to have a base level document which can be called a number of things, a 'charter', 'ground rules', 'golden rules', 'handbook', 'values', 'code of conduct' or whatever which everyone who works there signs up to. It can be quite simple, things like respect, professionalism, customer service, politeness, communication, team working are common topics. It's better if it can be developed 'bottom up' so the people actually doing the work decide what they want the rules to be, rather than it being imposed by the boss/senior management, and it's better if it's framed positively so things like 'always be polite and helpful to customers' rather than 'don't be rude to customers'. I like them to be as simple and as specific as possible - so things like 'be a good team mate' are too generic for me but 'help your colleagues if they ask for it' is better.
I think a lot of people are still getting their heads around working from home and what is and isn't acceptable, do you need to be available at all times, can you step away from your desk for 10 mins to do some housework/take the dog out, is it rude to IM/call your colleague just for a chat or to say good morning, is it rude NOT to occasionally IM/call your colleague to say good morning etc, so something focussing on this could be helpful. A good task for you could be to organise a variety of ways for people to feed in and discuss what they think could be the 'rules' and then synthesise this into something for your boss/senior management to endorse. You could do a survey monkey survey, a focus group/workshop on zoom, individual conversations etc. Just be careful if you know there are existing sensitivities/conflict not to accidentally stir it up as a result!
In terms of events, I agree you don't want anything people have to pay to participate in. At my work we've done a variety of things, some more cringey than others, some more successful than others. In terms of big 'group' things quizzes are a bit dull but always go down well (we have a 'winner hosts/quiz masters the next one' system), we've done a pretty embarrassing 'fancy dress party' (rules were you could only construct your fancy dress from materials already in your house, no spending money), a 'show off your pet' party (after one too many pet interruptions on zoom!), that one was fun, and a 'talent competition' where people showed off the various creative/crafty things they'd been doing in lockdown - didn't have to be a 'performance' as such, lots of people showed off their sewing/crochet/pics of their baking, and those people without a talent formed the audience/judges (thankfully, I'd have died if made to actually participate). As an introvert myself I have to say these big events suit the extroverts more, so we also do smaller, lower key things like small team 'tea party' gatherings where we just casually chat for 15 mins much as we would do in the staffroom at work, and also do 'coffee roulette' where we are paired up one on one with someone we don't know well to have again an informal/low key zoom chat over coffee. These are nice to keep up social connections and feel you still have more of a relationship. We have a team work whatsapp chat group which TBH gets a bit overwhelming at times so I tend to mute but is good for people to post silly memes, jokes, pics of their kids/pets/hobbies, gripes about lockdown etc and keeps that stuff out of email inboxes/serious meetings while giving people permission to also chat and have fun with their colleagues on occasion...