1. Yes, you can play over Discord. I do, in two separate campaigns, and it works wonderfully in the absence of actual table play. That said, we also supplement Discord with a virtual tabletop (in our case Tabletop Simulator).

2. No, you don't need to buy all the guides. You have a few options:

Cheapest Route: If you already have access to the content outside of D&D Beyond, you can use the homebrew tools to recreate what is missing for free. You may not publicly publish the homebrew, but it can be used by yourself.

Content Sharing: One of the subscription levels includes "content sharing." As long as one of the players has that subscription, that person can unlock content sharing for the entire campaign. That means that everyone in the campaign has access to what everyone has collectively purchased. You can divide the costs between the group, if everyone's willing. Please note that you only have access to shared content within that campaign. The compendiums can be seen at all times, but the digital toolset will only apply to characters you add to the campaign.

Piecemeal: Each book can be purchased a piece at a time. For example, the Player's Handbook is roughly $30 USD, but you can buy the compendium separately for $20. There are other options, like buying feats, magic items, or subclasses separately. For each piece you purchase, the total cost is reduced equivalently. If you purchase that $20 compendium, the whole book is reduced to $10. This also applies to the Legendary Bundle, though I'm not sure there if the reduced price is equal to the money spent.

Legendary Bundle: Expensive as a single purchase, but it is cheaper than buying all the books individually. Additionally, once purchased the discounted rate will apply to all future releases.

I started out as 'piecemeal,' because at first I was only interested in unlocking character creation tools to play around with. However, when I got the opportunity to DM a group of online friends, I went with the Legendary Bundle to complete my collection. I've also subscribed to unlock that content for my players. This put the responsibility of cost entirely on myself, but you could discuss with your group an arrangement to split that cost.

3. For starting a campaign, could you be more specific? Are you asking just about the technical aspects of it, or are you looking for something else? If you're DMing, you may want to ask in the DM's subforum for tips.