I’ll focus on the trust and connection part of your question - because a big part of being physically separate is occasionally coming together.
At my last job we were a tiny company (6 employees, all remote) and for building trust we had a quarterly company retreat. (We were fortunate enough to live within a few hours flight of each other). We’d hire an AirBNB and spend two or three days re-aligning around goals or working on some hack-a-thon style projects where we’d try to collaborate on something outside our routine intensely for a couple of days to shift the needle on something in the business. This time together was great for bonding.
At my current role (Culture Amp) I’m one remote worker in a mostly co-located group. We don’t do regular retreats like this as it’s easier for the handful of remote workers to visit the office. One thing we’ve done in the product group is look at the cost of keeping an employee in the office (rent, furniture, electricity etc) and given that budget to the remote engineers to make semi-regular trips to the office. It’s up to me to decide how to spend that budget, where to stay, how often I visit, when (the ask is at least one week every 3 months, but you can do more than that). I’ve really valued this - the trust to go and manage that myself is valuable. Whenever I’m in town I focus on face-to-face activities as much as possible, and we usually do a team lunch (or two!)
As for building trust and connection when you’re not physically together:
When you do a video call, be generous with your time at the start and at the end. In an office people often gather outside a meeting room a few minutes early or for a few minutes after and chat. Encourage people to start/join the video call early, and don’t end it as soon as the meeting is finished - you might get a few minutes to catch up
Clearly communicate expectations around availability. In an office you can go for a coffee and no-one questions it. When remote, if you go for a coffee and miss a Slack message, people wonder what you’re doing. Getting a coffee just like everyone else! It helps to post “AFK” (away from keyboard) statuses etc if you have a free flowing team chat.
At Culture Amp we use Donut (https://www.donut.com/) for a coffee roulette - it pairs you with another employee that you don’t normally interact with and encourages you to book a coffee or video call catchup with that person, purely social, for about 30 minutes. As a remote worker I really love these conversations and it helps me feel like I’m getting to know other people at work, even if I rarely see them in person.
I also like having some non-work-related Slack channels that make up for the lack of “watercooler” discussion when remote. We have “#club_music”, “#club_public_speaking”, “#club_cooking” and more. These all help the sense of social connection - it’s not directly related to work, but when you go to work with these people, even remotely, you now feel better connected with them and it feels more natural to collaborate.