Can keeping the grass round your house cut very short stop mosquitoes coming to your house?

No. But getting rid of standing water can certainly limit their breeding sites.
Unlikely to help. U can decrease a local mosquito population by eliminating breeding sites.Old tires, water catching toys, flower pots, barrels, bird baths, drainage ditches, etc are all sites of larval production.Depending on the area, public control measures may be required.Avoiding dusk or dawn activities & using loose long sleeved clothing helps. Repellants also help.