help understanding bulk email and spam

We have a client that insist on sending on bulk email....
(They also send attachments etc..)

We keep insisting for them to stop.
So they started sending via optonline...(Using the same address))

1- Now all other sites that we host at had no problems...
but thier site has crashed...many times...forms have not been working...
2- Their site has gotten "whitelisted"
3- They get an enormous amount of spam.
4- I had to keep added to thier bandwidth.

We know it's not optonline or our host issue. And because we do not have
total control...there can be many variables....

We know if we move to another host...the problem would still exist..
Correct?

What we suggested is they...
1- Do Bulk mail via an email marketing company via constant contact.
2- hire a programmer to set up a program to resolve this.
3- We do not want to resign the account...but we would like to move thier site to another url..
and keep thier email....and forms to this url. (Side note they would like to do a database for the forms..
and have the control of that.) We are not sure that is possible to set up.
4-get a dedicated server.

Can you suggest a simple explanation....or a site that can help me...
Understand this.

Thank you...