Be careful here when you say the word Tekkit, it is usually treated with contempt, bile, lipase and H2SO4 sulfuric acid vitriol sometimes here.
1.
Uranium as fuel has a set time limit of 10k reactor ticks, or 10k seconds, which gives that 2 h 46 min 40 s limit. If you have GregTech (which does not support Tekkit), it has Thorium fuel and Plutonium fuel which last longer. They were changed recently, but I remember their old values were 50k reactor ticks (13 h 53 min 20 s) and 20k reactor ticks (5 h 22 min 80 s) respectively. The values may change depending on server tick-rate.
2 & 3.
It is possible, however Uranium is not infinite without using some cheaty mod like EE2, as each uranium/thorium/plutonium cell has only a 25% chance of leaving a Nearly Depleted Uranium Cell, and there was (no longer is) a UU-recipe for Uranium. Thorium was centrifugable from Coal Dust IIRC but that has been removed. Plutonium requires Uranium to make, so that's also not renewable. You will also need Tin to craft the cells, however that is renewable through UU-matter, and Copper, which is also renewable through UU-matter, for the multicells. In case your design use Reflectors, which are a bad idea in general, you will also need Coal Dust, which is renewable through UU-matter.
Regarding automation: It is however you can't just use ComputerCraft without MiscPeripherals (I'm not sure if that's in Tekkit) as you need item automation since you will need to insert Uranium Cells.
Possible methods of item automation
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Buildcraft: Uranium, Tin and Copper to autocrafter of choice to produce cells. You may need a Diamond Pipe to control how much cells go where if your design uses a mixture of different cell types. After that, you need to fill up blank spaces in the reactor with the Reactor Platings (available in 3 varieties, Heat-Capacity and Containment) to prevent Buildcrash from inserting them into the wrong spaces. If you use Condensators, then too bad as Condensators are also bad ideas and Buildcrash won't know which slot to insert into. You will also need Emerald Pipes to extract the NDUCs. Additionally, the age of Buildcrash automation has been long past as it is pretty well-known for causing lag if you process a load of items due to item rendering stuff and if your system doesn't handle overflow.
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Vanilla (1.5+): When paired with a mod that supplies Autocrafters you can actually get the functionality of Buildcrash with a hopper chain at a slightly more expensive cost (5 iron per hopper) or you could hook up redstone with Droppers and ComputerCraft.
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Redpower: Also a well-used automation. RedPower has the downside of no standalone autocrafter, but IIRC Tekkit has TubeStuff which has a RedPower-compatible autocrafter. It will also cause issues with Condensators like previously said. RedPower automation used to be near flawless but Eloraam has disappeared and there are some bad crash bugs with pr6. It can also cause massive lag with the timer's lighting updates (but you can use ComputerCraft pulsers) and the re-rendering (or so have I heard) of the Filters with backlogs and stuff. Lag will also hit you big if you have lots of stuff flowing through tubes.
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Factorization: Factorization's Routers are actually quite awesome when paired with another automation mod as you can specify which slot to insert into.
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GregTech: Since you have Tekkit, no GT for you. However if you plan on ditching that modpack then GregTech has some pretty awesome automation when you look past the nerfs that is sadly under-documented. The Advanced Regulator and Advanced Buffers are actually quite awesome for this purpose. You will have to sacrifice your EU output as they require EU to do so, but quite a small amount, really. It also comes at an expense of no items visually flowing but that actually improves performance.
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Applied Energistics: Also not sure if in tekkit, but it is a fairly new mod that adds very awesome automation. Like GregTech no item visually flowing but that improves performance. It also has some visually cool blocks and models. It is rather expensive energy-wise to maintain though.
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MiscPeripherals: Again, not sure if in Tekkit but there are Inventory Turtles which can output into a specified slot though.