Happiness

Give it a massage at Hearthome City every day. Get about 15 of 1 stat-boosting item and give it to Eevee. If it says that it is already at the max, take it to Doctor footstep. Put Eevee in the front and talk to him. If it says " is a remarkable human and trainer blah blah blah"
Then level it up once during the night and it will evolve into Umbreon. (be carful. If you mess this up and do it during the day, it will be Espeon). Hope This Helped and gl

Yes to both.

Give it poffins it likes and massages helps alot

You could give Eevee a soothe bell or massages like they said but you don't have to trade Eevee all you have to do is level him up at night when he is happy (ONLY AT NIGHT). Also I can't remember to well but if have beatin the game and you get I think 9 or 10 ribbons with one Pokemon you can go to the resort area and get a spa treatment I'm prety sure it makes your Pokemon happy.
An easy way to get ribbons, there is a woman in Sunyshore city that will give you a ribbon every day for each day in the week.That should get you seven which the rest you can get from contests,oh yea you can also get one from beating the Pokemon league.
Hope I Helped.

Besides having Eevee hold a Soothe Bell, give Eevee massages to boost the happiness more quickly.

Yes it does help give help to give it a soothe bell.

Yes it does help to give it Stat raising items such as calcium, protein, iron, etc.

Once you think you have filled your Eevee's happiness, wait until night time and either level your Pokemon up through rare candy or experience gain. (Experience gain might be better in this occasion)

Also using your Eevee in battle as the first Pokemon in the first slot also makes your Pokemon feel wanted and needed, therefore raising it's happiness.

I hope this helps.