To repair craft with a busted engine you normally just need to send a ship with maintainance supplies out to them and transfer them across to the damaged ship. It will then use it's damage control rating to fix the engine. It will take twice the maintainance supplies to repair a part as are removed when a ship suffers a maintainace faliure to that part and is immediately repaired which you see in the event log.
If you are having frequent problems with ships running out of maintainance supplies to keep themselves running, you are either not placing enough engineering spaces on your classes or you are leaving them out for long periods without returning them for overhaul.
The alternative is to create tugs, ships equiped with a tractor beam and a high percentage of space dedicated to engines, to tow your damaged ships back to shipyards for repair.
Just took a look at your screenshot. What you have there isn't damaged ships, they are destroyed ships. Either you have an enemy running around blowing your ships up, or you are not sending ships back for resupply of maintainance supplies and refit and ther are suffering breakdowns until the entire ship fails. Are you putting military engines on your terraformers/colony ships? If not then you almost certainly have an enemy running around as commercial ships don't suffer maintainance faliures.
Terraformers
Terraforming takes a long time, do not expect to see results in less than multiple decades unless you have significantly researched the terraforming rate and are using many terraformers.
As with the asteroid miners you need to have a colony on the planets you want to terraform. If you are using terraforming ships, move them to the colony(don't use the orbit command as that doesn't actually move them to the planet in question). Setting up the terraforming is done via the enviroment tab on the colony screen. Select the gas you want to alter the level of, enter the target level, select the checkbox if you need to add gas to reach the target then hit "save atm".
In order to fully terraform a planet you need to do several things. First you need to heat up/cool down the planet to within your species temperature tolerance. All gasses add some (1/10th of their atmospheric pressure) warming effect, carbon dioxide and safe greenhouse gas add their full atmospheric pressure in addition. Anti greenhouse gas gives a cooling effect. Secondly you need to add/remove enough oxygen to place the oxygen pressure within your species tolerances. Thirdly you need to ensure that the percentage of oxygen is 30% or lower of the total atmospheric pressure. Finally you need to remove any dangerous gases such as chlorine.
If you are using terraforming installations their use is the same, you just need to move them to the colony via freighters.
Taken from the wiki
Transporting Installations
The facilities that can be transported are (requires 5 Cargo Holds unless otherwise stated):
Research Labs (100 holds in one go, or automatically broken down into parts)
Genetic Modification Centre (100 holds in one go, or automatically broken down into parts)
Maintenance Facilities
Construction Factory
Automated Mine
Mine
Mass Driver
Fuel Refinery
Ordnance Factory
Fighter Factory
Deep Space Tracking Station
PDC Component (1 cargo hold)
infrastructure (one half of a cargo hold)
Terraforming Installation (25 holds in one go, or automatically broken down into parts)
Ground Force Training Facility (100 holds in one go, or automatically broken down into parts) (ver. 5.50)
If you are following most peoples advice your freighters will have 5 cargo holds each and as such will take 5 trips to transport a complete terraforming installation. The easiest way to do so is to give the freighter orders something like;
move to earth
load terraforming installation
move to mars
unload terraforming installation
Then replace the 1 by the repeat orders button with 4 and hit the button which will give you a total of 5 trips. If your freighters have a different number of holds adjust the number of repeats accordingly.
Note that terraforming installations require there to be population to operate them which can be a problem for new colonies, so terraforming ships are normally recomended.
Colonising
Colonising is simply moving population from a source world to a target world in the same way as transporting anything else, just using colonist modules instead of cargo holds. However unless your target world has a colony cost of zero, population will start to die off as soon as they are dropped off. There are two ways to deal with this, either terraform the planet until it has a colony cost of zero, which takes considerable time, or shipping infrastructure via freighters to the colony in order to suport them. Infrastructure is built via the industry tab. The ammount of infrastructure required depends on the colony cost and can bee seen on the summary tab of the colony window.
Assuming you have suitable designs with commercial engines and cargo holds/colonists modules and give them sufficient funds to build them, your civilian shipping lines will handle most of the work keeping a colony growing. All planets create a certain ammount of infrastructure for free that the civilian freighters will ship to where it is needed, meaning you will only need to build more yourslef if the demand outstrips the supply. In order to get a colony started however you will need to ship the inital load of colonists/infrastructure yourself.