Which algorithm to use for percentage features in my DV and IV, in regression?
Having a mix of predictors doesn't matter for any form of regression, this will only change how you interpret the coefficients. What does matter, however, is the type/distribution of your Y variable

Case 1: Predict a continuous valued Y
a.Will using a Lasso regression suffice?

Regular OLS regression will work fine for this

b. How do I interpret the X-coefficient if X is standardized and is a
numeric value?

The interpretation of coefficients always follows a format like "for a 1 unit change in X, we expect an x-coefficient amount of change in Y, holding the other predictors constant"
Because you have standardized X, your unit is a standard deviation. So the interpretation will be "for a 1 standard deviation change in X, we expect an X-coefficient amount of change in Y..."

c. How do I interpret the X-coefficient if X is standardized and is a
%?

Same as above. You units are still standard deviations, despite it originally coming from a percentage

Case 2: Predict a %-ed valued Y, like % resource used.
a. Should I use Beta-Regression? If so which package in Python offers
this?

This is tricky. The typical recommendation is to use something like binomial logistic regression when your Y outcome is a percentage.

b. How do I interpret the X-coefficient if X is standardized and is a
numeric value?
c. How do I interpret the X-coefficient if X is standardized and is a
%?

Same as interpretations above. But if you use logistic regression, they are in the units of log odds. I would recommend reading up on logistic regression to get a deeper sense of how this works

If I am wrong in standardizing the Xs which are a % already , is it
fine to use these numbers as 0.30 for 30% so that they fall within the
range 0-1? So that means I do not standardize them, I will still
standardize the other numeric IVs.

Standardizing is perfectly fine for variables in regression, but like I said, it changes your interpretation as your unit is now a standard deviation

Final Aim for both cases 1 & 2:
To find the % of impact of IVs on Y. Eg: When X1 increases by 1 unit,
Y increases by 21%

If your Y is a percentage and you use something like OLS regression, then that is exactly how you would interpret the coefficients (for a 1 unit change in X1, Y changes by some percent)
I am using regression to analyze server data to find feature importance.
Some of my IVs (independent variables) or Xs are in percentages like % of time, % of cores, % of resource used, while others are in numbers like number of bytes, etc.
I standardized all my Xs with (X-X_mean)/X_stddev. (Am I wrong in doing so?)
Which algorithm should I use in Python in case my IVs are a mix of numeric and %s and I predict Y in the following cases:

Case 1: Predict a continuous valued Y
a.Will using a Lasso regression suffice?
b. How do I interpret the X-coefficient if X is standardized and is a
  numeric value?
c. How do I interpret the X-coefficient if X is standardized and is a
  %?
Case 2: Predict a %-ed valued Y, like "% resource used".
a. Should I use Beta-Regression? If so which package in Python offers
  this?
b. How do I interpret the X-coefficient if X is standardized and is a
  numeric value?
c. How do I interpret the X-coefficient if X is standardized and is a
  %?

If I am wrong in standardizing the Xs which are % already, is it fine to use these numbers as 0.30 for 30% so that they fall within the range 0-1? So that means I do not standardize them, I will still standardize the other numeric IVs.

Final Aim for both Cases 1 and 2:
To find the % of impact of IVs on Y. 
  e.g.: When X1 increases by 1 unit, Y increases by 21% 

I understand from other posts that we can NEVER add up all coefficients to a total of 100 to assess the % of impact of each and every IV on the DV. I hope I am correct in this regard.