Poor*
*This only applies to their starting lifestyle for adventurer league play. For a home game you can let them start with whatever they want. It's important to remember that adventurer's league is a specific interpretation of the rules for organized play.
More importantly though, this gets at how downtime works in Adv League. Downtime days in Adv League work as a specific form of currency. When you complete a part of an adventure you get 10 days. To spend those days you pay a specific amount (the living expenses for your desired lifestyle level). (ALPG 5,6)
"Practicing a Profession" is a specific way to spend your downtime. Using it this way doesn't cost you anything, and normally a character in town can do this at a comfortable lifestyle (if trained in performance and you choose to use it, you can do so at wealthy). However, there is another option for this particular item. (PHB 187)
When you spend time "practicing a profession" and want to do so in the wilderness, you can do it and be considered poor. If you are trained in survival you can do it and be considered comfortable. However, neither of these things have anything to do with the starting lifestyle of someone trained in survival. (PHB 159)
It's also important to note that those are the default starting lifestyle expenses. You are free to go up or down if you want. If you wan to go down a level, simply spend a day paying at the lower level. If you want to go up a level, spend 30 days paying at the higher level. For whatever reason, they have decided that the Outlander should start at poor, this is obviously not respective of the fact that the background comes with training in survival, but merely a decision made by WOTC's organized play team.
First off RAW gives the referee the authority to decide whether downtime activities are required.
Between adventures, the DM might ask you what your character is
doing during his or her downtime
In the PHB on page 187, we have a rule requiring the player to choose a lifestyle for their character and pay the appropriate expenses.
Between adventures, you choose a particular quality of life and pay
the cost of maintaining that lifestyle, as described in chapter 5.
Living a particular lifestyle doesn't have a huge effect on your
character, but your lifestyle can affect the way other individuals and
groups react to you. For example, when you lead an aristocratic
lifestyle, it might be easier for you to influence the nobles of the
city than if you live in poverty.
If your referee keeps strict track of time between adventure this expense can add up. If the income from adventuring doesn't cover it then the player may have to have their character do something in between adventures i.e. downtime activities to cover their expenses.
Now the lowest lifestyle is Wretched which cost nothing per day. However the description of Wretched is.
You live in inhumane conditions. With no place to call home, you
shelter wherever you can, sneaking into barns, huddling in old crates,
and relying on the goad graces of people better off than you. A
wretched lifestyle presents abundant dangers. Violence, disease, and
hunger follow you wherever you go. Other wretched people covet your
armor, weapons, and adventuring gear, which represent a fortune by
their standards. You are beneath the notice of most people.
While there no direct mechanical consequence of living a Wretched lifestyle, the part highlighted in bold give the referee ample authority per RAW to make the character's life a hell between adventures. The character would win at first but likely be worn down by the incessant attacks and interruptions.
So while engaging in downtime activities is not specifically required, it would not be smart not to at least do something to insure a decent lifestyle..
Best Answer
Practicing a Profession is one of the ways to spend your downtime. Specifically, it's a way to spend your downtime without paying a fee (provided you want to maintain a comfortable lifestyle). If you practice a profession in town, you can lead a modest, comfortable or wealthy one (depending on circumstance), if you do so out of town, you can lead a poor one (unless proficient in survival which allows comfortable).
If you don't want to work, and want to spend your downtime doing something else (or nothing at all), you must pay for whatever lifestyle you want to afford. So if you don't want to pay, you'll lead a Wretched lifestyle, with the benefits and penalties that come with that (and trust me, there will be serious penalties to doing this longterm).
Lifestyle expenses are listed in the chapter 5 of both the PHB and BD&D rules (p 157 of the PHB, p 53 of BD&D). They are also described there. The rules are effectively, you are considered to live the lifestyle you pay for (though the Adventurer's league is a bit more strict, to drop you only have to spend 1 night living the lower lifestyle, but to increase you must spend 30 days paying for it).
As far as living in the wild without practicing a profession, you'll still need to pay for a lifestyle then, because "practicing a profession" in the wild represents the things you need to do to survive, if you're not spending time catching food, maintaining your tools and habitat, you're going to have to pay someone else to do so if you want to maintain a specific standard of living. Otherwise you're living conditions are considered wretched and you would be subject to whatever conditions the DM wills on you (likely disease and bandits on a frequent basis).