Do I need all three?
Yes. SPF and DKIM prove a mail is genuine; DMARC tells receivers to act when it isn't, and is the only one that stops someone forging your exact address.
Will p=reject stop my own email?
It can, if a service sends as you without being in SPF or signing with DKIM. That's why you start at p=none and read the reports first.
Why do my password reset emails go to spam?
Usually missing or failing SPF, DKIM or DMARC, or sending from a shared domain. Send from your own domain through a transactional provider with all three set up.