Yes, you can initiate a grapple and pin in the same round.
That is exactly what Greater Grapple is for, it explicitly allows you to make an additional check as a move action -- and the wording of grapple says nothing about having to wait until the next round to use any of the special grapple actions.
If you're worried about the word maintain, that seems to just be a poor choice of words on the designer's part. Greater Grapple specifically states that you may make such checks twice in one round; and the rules for maintain make it clear that this is the only type of offensive check that you make in a grapple. The various actions such as pinning/moving/etc only occur on a successful check to maintain:
Once you are grappling an opponent, a successful check allows you to continue grappling the foe, and also allows you to perform one of the following actions (as part of the standard action spent to maintain the grapple).
It can, or the cutting part wouldn't work.
The Pathfinder rules are based on a list of things characters can normally do, such as using skills, or attacking. This list is not absolute. A spellcaster can also cast spells, or you can't use diplomacy on someone trying to kill you at the same time. However, any deviation from the general set of possibilities needs to be explicitly mentioned in the rules.
Swallow Whole makes no mention of any sort of restrictions on the character, apart from him being grappled. Of course, being grappled itself imposes restrictions on what you can do, and the grappling rules are clear on what you can do if you are grappled, including
Instead of attempting to break or reverse the grapple, you can take any action that doesn’t require two hands to perform, such as cast a spell or make an attack or full attack with a light or one-handed weapon against any creature within your reach, including the creature that is grappling you.
Also take note about the description of cutting yourself free from Swallow Whole:
A swallowed creature can try to cut its way free with any light slashing or piercing weapon (the amount of cutting damage required to get free is equal to 1/10 the creature’s total hit points), or it can just try to escape the grapple. The Armor Class of the interior of a creature that swallows whole is normally 10 + 1/2 its natural armor bonus, with no modifiers for size or Dexterity.
This is not a special type of action only available to swallowed characters (Note how it doesn't say things like "as a standard action"). It basically just states that instead of just hacking away at the monsters insides, you can use some weapons (i.e. light slashing or piercing ones) to cut more precisely, in order to cut yourself free.
Best Answer
To the first question. Yes, "If you're able to make multiple attacks with the Attack action, this attack replaces one of them." (PHB 195) So yes you can attack again if you grapple with your first action. You are not limited to whom you attack, so anyone in reach including the grappled opponent is fine.
Your opponent is only frozen in place. That is the only restriction. He can attack you or anyone else if they are within his reach.
Advantage and Disadvantage work as normal, but nothing special is added due to the grapple.
There are not grappling specific attacks at this time, unless you have a specific spell or class feature that grants them. For instance the Grappler feat adds additional things to do with the Grappled condition, such as adding restrained.
So basically, as a part of your attack action, you can grab someone, all this does is set their speed to 0. That's pretty much it. There are other things you can do, but it's largely dependent on having class features or feats to support it. The WOTC forums have a lot more information in the Grappler's Guide that Ruut pointed to in his answer.