Your post is not clear so I've done my best with this answer.
The inability to get the chain to go onto either the largest or smallest sprocket (at the rear) is not dependent on the wheel. Adjustment of the rear derailleur limit screws and/or the cable tension is required.
With a 622-13 wheel (rim) you will not be able to fit wider tyres (wider than 28-622) safely.
Sheldon (scroll to 3/4 down) suggests the ratio of tyre width to (internal) rim width (13mm in the OP's case) should be no more than 2:1 and I think that's a good rule of thumb (although note last sentence in quote below).
"Although you can use practically any tire/rim combination that shares the same bead-seat diameter it is unwise to use widely disparate sizes. If you use a very narrow tire on a wide rim, you risk pinch flats and rim damage from road hazards. If you use a very wide tire on a narrow rim, you risk sidewall or rim failure. This combination [also] causes very sloppy handling at low speeds. Many cyclists exceed the recommended widths with no problem."