Yes, grapples and shoves count as melee attacks for the Mobile feat, according to RAW
The third benefit of the Mobile feat says (PHB, p. 168; emphasis mine):
When you make a melee attack against a creature, you don’t provoke opportunity attacks from that creature for the rest of the turn, whether you hit or not.
Thus, it only triggers this effect on a melee attack and does not care if that melee attack results in a hit or not.
Grappling and shoving are melee attacks
The rules for grappling say:
When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple.
The rules for shoving say:
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you.
Rules designer Jeremy Crawford has also made an unofficial ruling to the same effect in this tweet:
An attack involves an attack roll or doing something that the rules call an attack, like grappling or shoving.
So, grappling and shoving are are defined as attacks in the rules and designer clarification has verified that (many times in fact). Thus, they qualify as melee attacks for this part of the mobile feat.
Grapples/Shoves cannot hit, but this technically does not disqualify them
We know that grappling and shoving are attacks. However, they are unusual attacks because they do not use an attack roll.1 And because of that they cannot hit or miss - only succeed or fail (see Does grappling count as a hit?).
This is important because Mobile says that the effect triggers "whether you hit or not".
Despite the fact that they cannot hit or miss, technically the wording still leaves room for grapples/shoves to qualify. If it had said "whether you hit or miss" then you would not be able to use a grapple. However, it does not say that, it says "hit or not" and technically a grapple attack will always not hit because it cannot hit (it can succeed or fail). Thus, they still work via a strict RAW reading of the language.2
1 Because there is some confusion on the matter, the fact that shoving/grappling does not involve an attack roll does not in any way make them not an attack. Most attacks involve an attack roll; grapples and shoves are unusual in the fact that they don't involve an attack roll, but they are still attacks. As Jeremy Crawford once again clarified in an unofficial tweet:
The grapple option in the Player's Handbook is an attack, but it uses an ability check in place of an attack roll.
2 This kind of weirdness in the language seems to indicate that the feat was written without consideration for the fact that melee attacks that do not involve attack rolls do exist (they are in fact extremely rare). This seems more likely than an attempt to specifically exclude grapples/shoves. In any case, the language does not actually exclude them so my logic stands regardless of intent.
The targeted enemy can't make opportunity attacks
As you've quoted, the relevant part of the Swashbuckler rogue's Fancy Footwork feature description says (SCAG, p. 135; XGtE, p. 47):
During your turn, if you make a melee attack against a creature, that
creature can’t make opportunity attacks against you for the rest of
your turn.
The only requirement for the feature is that you "make a melee attack" against a creature on your turn; it doesn't specify that the attack has to hit (or miss), so it doesn't. Therefore, if you make a melee attack against a creature on your turn, then that creature can't make opportunity attacks against you for the rest of that turn. That's all there is to it.
Rules designer Jeremy Crawford reiterated this fact in an unofficial tweet from May 2016:
does the Fancy Footwork feature of the Swashbuckler rogue require the melee attack to be successful?
Fancy Footwork works whether or not your attack hits.
This tweet just repeats what's already self-evident from the feature description: you just have to make the attack - you don't have to hit.
As a sidenote, Fancy Footwork doesn't even require an attack roll, since it doesn't mention hitting or missing; it just says you need to make a melee attack against a creature. This means that even trying to grapple or shove a creature is enough to qualify for the benefit of Fancy Footwork, since both shoves and grapples are described as "special melee attacks".
This fact is also reinforced in a Q&A about the interaction of the third benefit of the Mobile feat, which works similarly (as you note): Does a Grapple or Shove count as an attack for the third benefit of the Mobile feat?
Best Answer
The feat doesn't limit the amount of attacks you can make. It only states that whenever you make a melee attack against a creature, you don't provoke opportunity attacks from that creature for the rest of the turn. In fact, multiple attacks work quite well with this: if you are adjacent to two enemies, you can make one attack against each and then leave their reach without provoking an opportunity attack from either!
In most cases, even if you have Extra Attack, you will only take at most one Attack action during your turn. Extra Attack doesn't grant you additional Attack actions, but increases the amount of attacks you can make whenever you take said action. Note that Mobile feat's bonus triggers off any melee attack, including melee spell attacks, opportunity attacks and other attacks made without using the Attack action.
The distinction between the Attack action and an attack as a game mechanic is notable as there are features that are triggered by taking the Attack action as well as features that are triggered by making attacks --- see this question and its answers for more information about the distinction.