Constrict, Rake and Rend are not natural attacks. They are special abilities that, under certain circumstances, enhance natural attacks. Improved Natural Attack improves the damage of one of your natural attacks (per feat), not one of your special abilities.
Rake and Savage Grapple are pretty similar in effect. Arguably, they could stack, but Savage Grapple is essentially a better effect than Rake (or a better version).
It would be like taking Weapon Focus (Cleave). Cleave isn't a weapon (or an attack), it's a feat that can affect your attack with a weapon.
You could however, have Weapon Focus ([Grapple or Unarmed Strike or Ray]), but these are called out specifically as non-weapon attacks that can be "focused".
Some confusion comes in from the usage of the term "Special Attack" in the monster listings. They have "Special Attacks and Special Qualities"
Many creatures have unusual abilities. A monster entry breaks these abilities into special attacks and special qualities.
However, this is different than what is listed under the combat section as "Special Attacks", such as grapple or bull rush. Specifically, a Special Ability is either extraordinary (Ex), spell-like (Sp), or supernatural (Su). This is where you find Constrict and Rake. Rend is only found in the monster entries, like the troll, but it follows the same format and is listed as
Rend (Ex)
Signifying it is a Special Ability, specifically an Extraordinary Ability.
Two-Weapon Fighting: When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use
a bonus action to attack with a different light melee weapon that
you're holding in the other hand. You don't add your ability modifier
to the damage of the bonus attack, unless that modifier is negative.
In D&D 5e there is no Two Weapon Fighting Style in the way that many other RPG define fighting styles.
Instead you are eligible to make another attack with a off-hand weapon if you met the requirements. In this case you have a light melee weapon in primary hand and a light melee weapon in the off-hand. This makes the character eligible to use their bonus action to make another attack with the condition that your ability modifiers doesn't apply to damage unless it is negative.
If the character is grasping two weapons but doesn't meet the criteria above then he can not take a bonus action to make a off-hand weapon attack per the above rule. Otherwise holding two weapons doesn't have any other effect.
As for the specific tactic the Rogue character use. You have to keep in mind that D&D 5e does not detail which hand is dominant for a PC. Many RPGs, for example GURPS, note that character will have a dominant hand and penalizes attack made with a weapon in the off hand. D&D 5e doesn't deal with this. Hence if you are grasping two weapons, you can do anything that you are eligible to do with either weapon. Including as stated in your example, attack with a Rapier in round one and with the staff in round two. In both rounds the character takes only the action he is eligible for except he using a different weapon held in a different hand.
I will note that because both hands are occupied then both weapons are considered gripped by only one hand. This may be important for specific weapons.
Best Answer
Yes
(Player’s Handbook II pg. 84)
There are no requirements or limitations on “both of the weapons” aside from there being two of them. If bludgeoning weapons could not be used with this feat, it would say so.
The name of a feat has no bearing on how it functions. Two-Weapon Rend uses “rend” because the mechanics are similar to the rend attack of certain monsters. So in this context it’s a reference to the mechanical term for extra damage for hitting with both weapons, not to the usual English-language definition of the word. But even if that weren’t the case, names serve as identifiers only—always look to the actual rules text for how things work.