1) Is there a source that a mezuzah should be uncovered so that you can see the klaf or the shin on the outside?
2) A door that is in used by two people (eg. one tenant lives upstairs, the other downstairs, and there is a door to get from one to the other) – who is obligated to pay for the mezuza?
1 Shulchan Aruch Y:D 286:5
2 If they both use the door as an entranceway, i.e. the door leads to an entrance way which leads to doors of both apartments, they would both be responsible, and should split the expense.
From your question is sounds like for one it is an exit and for another it is an entrance. If this is the case the obligation is on the one who this doorway functions as an entrance for.