B737-800 Captain reported on initial climb a landing gear issue corrected via QRH; but further reverser warnings resulted in expeditious handling return to departure airport.

Narrative

While operating flight from ZZZZ airport; gate departure; taxi and takeoff were normal. After lift-off; on initial climb landing gear did not retract. As pilot flying; I requested the First Officer tell ATC we needed a lower altitude level off and a heading away from weather as we accessed the problem and addressed the QRH. Completing the QRH procedure; step X directs the crew to open and close the manual gear extension access door." Upon completion of this step; the landing gear operated normally and retracted as prescribed. While completing the QRH for the landing gear disagree; it was noted that the "reverser" lights on the overhead panel were illuminated for engine 1 and 2. We completed the "reverser" QRH procedure and the lights remained illuminated. Although flight characteristics were normal and cabin crew reported visual inspection of the reversers to be normal; with compounding issues and communications to Dispatch and Maintenance Control limited; I made the command decision to return to the departure station of ZZZZ. Approximately 10;000 lbs over landing weight; we [requested priority handling] and completed the appropriate overweight landing checklist and the non-routine landing considerations checklist. Weather at the airport was VFR with favorable winds and the runway landing distance was reported. We requested safety vehicles be present as a conservative precaution and asked they inspect the brakes after landing. With operations normal; flight concluded at the gate without issue and at no time was safety compromised."