Let us evaluate all the given statements:

• $\alpha$-carbon of $\alpha$-amino acid is asymmetric except for glycine: This is a correct statement. In all standard $\alpha$-amino acids except glycine, the $\alpha$-carbon is attached to four different groups (an amino group, a carboxyl group, a hydrogen atom, and a variable R group), making it chiral or asymmetric. In glycine, the R group is a hydrogen atom, making the $\alpha$-carbon achiral.

• Some proteins of human body are found in L-form: This is an incorrect statement. All naturally occurring proteins in the human body are composed exclusively of L-$\alpha$-amino acids, not just 'some'.

• The human body can synthesize all proteins it needs: This is a correct statement. The human body synthesizes all its required specific proteins (enzymes, hormones, structural proteins, etc.) through the process of translation, provided it has the necessary pool of amino acids (both essential, which must be obtained from the diet, and non-essential, which the body can synthesize).

• At $\text{pH} = 7$, both amino and carboxylic groups exist in ionized form: This is a correct statement. At physiological $\text{pH}$ (around $7$), amino acids exist as zwitterions (dipolar ions). The acidic carboxylic group loses a proton to form a carboxylate ion ($-\text{COO}^-$), and the basic amino group accepts a proton to form a substituted ammonium ion ($-\text{NH}_3^+$).