Yes, you would need to specify an accurate weight in the Shipping Options screen for each product. The weight will not matter for US shipments, since you have configured a Flat rate for those.
To add $2 handling to ALL orders, you would to to Admin > Shipping, set a Handling charge as 2.00 Flat, then click next to reach Shipping Rules. This will be applied to ALL orders, so you'll need to adjust your US Flat rate to "uncombine" the Handling from that rate.
Select the Rule for your US-only Flat rate, then click Delete to remove the Rule. Use the Shipping Rate Calculator drop down menu and choose Flat. Enter in the new amount of $10.95 and then leave the Description blank or call it something like "US Flat Rate". Then use the Country drop down to select the United States. After that click Submit to save the rule.