How to handle donors that pay by wire transfer
I copy and paste your questions in slightly different order and intersperse them with my responses in an attempt to provide a clear answer:


Should I complete reconsider the usage of memberships for this type of setup entirely and do something different?



  
    
      Since, as you state, the entire family gets benefits as a result of a single donor making a donation, using memberships makes sense to me.
    
  



How can I automatically renew these memberships so they are extended but waiting for payment?



  
    
      if you were to use a payment processor like iATS who offers ACH capabilities and who works specifically with non-profit organizations and has developers who are dedicated to the development and maintenance of their CiviCRM extension, you might actually end up saving money since you would be able to set up recurring membership payments directly through Civi on whatever schedule your donors prefer (or your donors would be able to do so themselves online), and you would free up staff time to do other things than to re-enter financial information for payments handled outside of Civi.
    
  



How can I enter multiple payments for donors who split up their payments (as rediculous as it sounds, I have one who pays €2,17 every quarter), or for those who pay more but in multiple instances?



  
    
      Ah, if you want to speak of the ridiculous, how about a donor that asks to split a $18 contribution to 6 different funds? Joking aside, if you still choose not to use a payment processor, then as Eli suggests, using pledges to be paid in installments is the method that I am most familiar with, but this means maintaining memberships separate of financials which may not be the best process for you. 
    
  



How can I tell which households still need to pay (part) of their fee?



  
    
      Again, the use of pledges is a process that I am more familiar with than the use of membership contributions, but if you continue to manually record these contributions in the back-office, you can enter a membership contribution with a status of "partially paid" or "pending" (in the case that no payment was received upon initial creation of the membership). 
      
      To search for pledges with balances still due, Contributions > Pledges > Find Pledges and select the pledge status of Pending, In Progress, Overdue, and Partially paid 
      
      To search for memberships contributions that are either pending or partially paid, you can use Contributions > Find Contributions, select the relevant pending statuses, and select the financial types you use for those membership contributions.
    
  



How can I send an e-mail to donors when the e-mail address is on (one of the) household members and not on the household itself?



  
    
      For this, you would either need to have the email address of the primary member on the household record, or you would need to maintain the memberships on the individual record (and related contacts would be provided membership according to the membership type configuration). The latter method is the one we recommend to clients given that when members login, they typically do so with an an individual login, not a household login - if you eventually open up the option for online membership signup, you will then have a mix of household vs. individual memberships that may become difficult to manage and report on consistently. Detailed instruction on how you would set up memberships to manage this process can be found at the following link: https://docs.civicrm.org/user/en/stable/membership/defining-memberships/#membership-types
      
      If the purpose of your email is to remind members of renewal, you can use scheduled reminders - documentation for setting these up on memberships can be found at the following link: 
      https://docs.civicrm.org/user/en/stable/email/scheduled-reminders/#chasing-members-who-have-not-sent-membership-payments
    
  


So, as you can see, there is quite a bit of research to be done, but the best way to figure out the best way to track this information is to learn through testing to see if the end result provides you with easily obtained information through reports that already exist within CiviCRM.

Good luck,
Tamar