Would this result in a redundant internet connection?
Would the working connection overtake a network stream if the other connection fails?
It would, but it will require configuration.
This is called fail safe connection.
Since your question is not about specific setup or install I would say that there should be process which monitors connectivity on the interface and if it detects failure (unable to ping or retrieve control content from external resource) it brings second interface up and does all needed reconfiguration so that all traffic goes via second network interface and this is transparent to the system and user. Optionally it can switch back to first interface if it goes up. (Let's say second interface connection provider is more expensive, so you want to use it only if first goes down)
Is it possible to use this two connections as one as long as both of
them are working (Bandwidth of connection1 + bandwidth of
connection2)?
Again, answer is yes, but it requires special configuration.
This is called load balancing connection.
Here all traffic is being sent to destination via either first or second interface. It is possible to "stick" particular protocol or addresses to particular interface (for example, router all SMTP via first interface). You should be careful here, because there can be session problems while using this technique. Let's say you are browsing you bank account and connection was established from first interface with address A. Now bank server expects all packets from you from address A. If your load balancer will decide to send next packet from second interface which uses address B than you, most likely, will get a session error on the bank web page. But it is possible to route entire session to particular interface.
These two technologies often implemented to the network routers with "easy-to-configure" settings and more professional equipment.
I should mention some switches use "Port aggregation" as well. You can combine few physical connections between switches and they will become "one big connection" with combined capacity of its members. (Three 1GB/s connections will give a 3 GB/s connection).
