1. Yes you can say 'I'm going to check that'.
2. Check up - a check up is usually used for a standard appointment at the doctors, dentist or opticians. You say 'I'm going for a check up' to see if everything is OK. check up can also be used like you can check up on someone to see if they are OK. Like if a friend is I'll, you would check up on the to see if they are OK or need anything.
Check out is used for checking out/leaving a hotel for example. Or you can 'check out' a person - seeing how attractive a person is. Check out is also used like: 'his story didn't check out ' which means it was a lie.
Check up - checking everything is OK.
Check out - leaving a hotel, observing an attractive person, unconvering of something said is true or a lie .
I hope this helps.