No. You cannot pick specific checks or deposits before LedgerSync pulls them from the bank. LedgerSync pulls all the check and deposit images for the period it fetches, and you sort through them once they are in LedgerSync.
No. A refresh fetches images for all of the connection's active accounts, for the dates being pulled. There is no option to choose individual checks or deposits, or to pull only checks or only deposits.
Once the images are in LedgerSync, go to Bank Connections > Checks. There you can:
Every image LedgerSync pulls counts for billing, whether it is a check or a deposit.
Yes, for a Statements & Checks connection. On the connection, click Refresh data, choose the From and To dates in the date picker, then click Submit. LedgerSync pulls the check and deposit images for that period.