Absorption of amino acids can be affected by the foods with which you take them, and by other amino acids. Amino acids are classified into three general groups (acidic, basic and neutral) and there are specific transporters for each group in the small intestine. A large amount of one amino acid may compete with a smaller amount of another in the same group. Unfortunately, there is not enough research in humans to know exactly at what dosage this can be a problem. Complicating matters further, there are situations in which ingestion of certain amino acids may aid the absorption or physiologic action of others.
This lack of clarity is likely why amino acid products instruct you to take the supplement without food, playing it safe that there will not be competition with other amino acids or nutrients for absorption, although you may also be forgoing the potential for improved absorption or action. Until more is known, you are probably best off sticking with the instructions on the label.
Also be aware that eating food can slow digestion and may delay the absorption of your amino acid supplement. This could be problematic if you seek earlier absorption to promote muscle repair during or immediately after a work-out.
Regarding your difficulty swallowing capsules, try this trick: hold the capsule in your mouth while you take a drink of water, then, swirl the capsule around in your mouth with the water, allowing the entire surface of the capsule to get wet, and then swallow the water and capsule together. This may prevent the capsule from "sticking" on the way down.
If you take the amino acid arginine, note that it may cause stomach upset (as discussed in the Concerns and Cautions section of the L-Arginine Supplements Product Review).