You can, being knocked prone doesn't stop you from being grappled
First of all, welcome to Stack Exchange!
As per grapple rules, emphasis mine:
The target of your grapple must be no more than one size larger than you and must be within your reach. Using at least one free hand, you try to seize the target by making a grapple check [...]
No limitation on character state stated. To grapple a prone opponent, simply declare your grapple and roll the appropriate check.
However, the grapple's target isn't under any additional penalties to escape your grapple, as escaping a grapple doesn't require a movement action. Again, as per rules:
Escaping a Grapple: A Grappled creature can use its action to escape. To do so, it must succeed on a Strength (Athletics) or Dexterity (Acrobatics) check contested by your Strength (Athletics) check.
As for your second question, which could possibly be its own question entirely:
Grappling does not apply any negative effects on the grappling character
There's no rules stating otherwise that I can find.
Extra: A prone and grappled creature retains the following effect from Prone:
An Attack roll against the creature has advantage if the attacker is within 5 feet of the creature. Otherwise, the Attack roll has disadvantage.