Generally, yes. Hotspot Shield uses OAuth 2.0, meaning they receive a token, not your Google password. However, if you lose access to your Google account, you lose access to Hotspot Shield.
You are permanently locked out. Because Hotspot Shield prioritizes privacy, they cannot manually change your email via support chat without you logging in first. You will need to create a brand new account. hotspot shield login
If you’ve forgotten your password and the reset email isn’t arriving: Generally, yes