Have you explored intercom.io? Based on what you have listed, it can fulfill all the use cases but yes depending on your size, it can be an expensive proposition beyond a certain customer base.

This is what I have used to fulfill the above use cases -
1. Lead capture - through different channels automated using zapier, integromat, automate.io - basically spread your use cases across different automation and then add the lead data to a single crm automation system. I prefer customer.io and sendx.io
2. I use customer.io to basically use it for existing customer or atleast who have already shown interest
3. I use sendx.io for cold mail, drip campaigns to them
4. Not using live chat at this point. But if I use this, again would prefer a tool that have a webhook which I can then use to send the data to either senx.io or customer.io.

Hope the above helps.